#390409

order	512
nilpotency class	3
isoclinism	isoclinic class of #30871
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2675
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14572
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1322 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1322 conjugacy classes of subgroups — too many to draw. Only its 232 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)(3,14,19,41)(4,20)(5,10,29,40)(6,11)(7,13)(8,17,33,18)(9,24)(12,30)(15,31,45,59)(16,37)(21,34,48,58)(22,51,54,49)(23,36)(25,42)(26,55)(27,38,28,39)(32,46)(35,52,61,53)(43,47)(44,57)(50,60)(56,62)(63,64);(1,3)(2,8)(4,21)(5,26)(6,18)(7,19)(9,34)(10,37)(11,41)(12,14)(13,33)(15,23)(16,27)(17,30)(20,31)(22,44)(24,45)(25,48)(28,47)(29,46)(32,38)(35,57)(36,58)(39,55)(40,43)(42,59)(49,62)(50,52)(51,60)(53,56)(54,63)(61,64);(1,4,7,25)(2,9,13,36)(3,15)(5,22,29,54)(6,23,30,24)(8,31)(10,35,40,61)(11,20,12,42)(14,34)(16,44)(17,21)(18,48)(19,45)(26,56)(27,52,28,53)(32,57)(33,59)(37,62)(38,51,39,49)(41,58)(43,60)(46,50)(47,63)(55,64);(1,5)(2,10)(3,16)(4,22)(6,28)(7,29)(8,32)(9,35)(11,39)(12,38)(13,40)(14,43)(15,44)(17,46)(18,26)(19,47)(20,49)(21,50)(23,53)(24,52)(25,54)(27,30)(31,57)(33,55)(34,60)(36,61)(37,41)(42,51)(45,63)(48,56)(58,62)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 297605);