#6370739

order	512
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#120
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#379
prime radical	#63103	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

16913 conjugacy classes of subgroups; only the 1808 normal subgroups are shown.

normal subgroups only

This group has 16913 conjugacy classes of subgroups — too many to draw. Only its 1808 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,6,12,7,11,24,33)(3,13,17,34,16,8,23,27)(4,9,20,29,19,30,44,54)(5,10,22,32,21,31,46,55)(14,25,39,49,38,35,45,53)(15,36,41,56,40,26,47,50)(18,28,43,52,42,51,60,63)(37,48,59,62,58,57,61,64);(1,3)(2,8)(4,14)(5,15)(6,23)(7,16)(9,25)(10,26)(11,13)(12,34)(17,24)(18,37)(19,38)(20,45)(21,40)(22,47)(27,33)(28,48)(29,53)(30,35)(31,36)(32,56)(39,44)(41,46)(42,58)(43,61)(49,54)(50,55)(51,57)(52,64)(59,60)(62,63);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,25)(10,28)(11,30)(12,29)(13,35)(15,37)(16,38)(17,39)(21,42)(22,43)(23,45)(24,44)(26,48)(27,49)(31,51)(32,52)(33,54)(34,53)(36,57)(40,58)(41,59)(46,60)(47,61)(50,62)(55,63)(56,64);(1,5)(2,10)(3,15)(4,18)(6,22)(7,21)(8,26)(9,28)(11,31)(12,32)(13,36)(14,37)(16,40)(17,41)(19,42)(20,43)(23,47)(24,46)(25,48)(27,50)(29,52)(30,51)(33,55)(34,56)(35,57)(38,58)(39,59)(44,60)(45,61)(49,62)(53,64)(54,63);(5,21)(10,31)(15,40)(18,42)(22,46)(26,36)(28,51)(32,55)(37,58)(41,47)(43,60)(48,57)(50,56)(52,63)(59,61)(62,64)

Polycyclic presentation:

2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^b=gi,h^b=hi,f^c=fi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72342755, #72342758, #72342778
maximal subgroup of	#72303182, #72303199, #72303219, …
maximal normal subgroup of	#72303182, #72303199, #72303219, …
maximal quotient of	D₄×#1791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277935);