#392137

order	512
nilpotency class	3
isoclinism	isoclinic class of #392135
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14679
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×2₋¹⁺⁶	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

15639 conjugacy classes of subgroups; only the 688 normal subgroups are shown.

normal subgroups only

This group has 15639 conjugacy classes of subgroups — too many to draw. Only its 688 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^b=gi,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262950, #72263037, #72263043, …
maximal normal subgroup of	#72262950, #72263037, #72263043, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299333);