#124262

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1519
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7572
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

475 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 64 points:

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

Polycyclic presentation:

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

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, 31458);