#124286

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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	#7573
prime radical	C₂⁴	#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 329 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)(3,13,16,34,17,39,26,54)(4,18,20,9)(5,22,24,10)(6,12)(7,11)(8,15,29,45,30,46,31,23)(14,40,42,58,44,38,43,61)(19,32)(21,49,53,33)(25,37,55,35)(27,36)(28,41,56,50,47,62,57,63)(48,52)(51,59)(60,64);(1,3)(2,8,11,30)(4,14)(5,23)(6,26)(7,17)(9,28,18,47)(10,34,22,54)(12,29,36,31)(13,37,39,35)(15,25)(16,27)(19,50)(20,44)(21,42)(24,45)(32,58,48,61)(33,57,49,56)(38,60,40,59)(41,51)(43,53)(46,55)(52,63)(62,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,28)(10,32)(11,18)(12,33)(13,38)(15,41)(16,43)(17,44)(22,48)(23,50)(24,52)(25,51)(26,42)(27,53)(29,57)(30,47)(31,56)(34,58)(35,59)(36,49)(37,60)(39,40)(45,63)(46,62)(54,61)(55,64)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,d^b=dg,e^b=efgi,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, 31482);