#125565

order	512
nilpotency class	4
isoclinism	isoclinic class of #125496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,4,7,5,6,3,8)(9,10)(11,22,25,49)(12,28,30,18)(13,33,35,47)(14,20)(15,38)(16,21)(17,24)(19,36,45,37)(23,50)(26,44,27,57)(29,61)(31,60,32,43)(34,46)(39,51,58,71)(40,55)(41,56)(42,63)(48,54)(52,70)(53,68)(59,64)(62,67,72,69)(65,66);(2,6)(3,4)(9,11,16,25)(10,17,21,41)(12,23,30,54)(13,34,35,55)(14,27,15,26)(18,39,28,58)(19,44,45,57)(20,46,38,40)(22,47,49,33)(24,36,56,37)(29,62,65,72)(31,53,32,52)(42,68,59,70)(43,69,60,67)(48,61,50,66)(51,63,71,64);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,23)(13,29)(14,31)(15,32)(16,30)(17,39)(19,42)(20,43)(21,28)(22,48)(24,51)(25,54)(26,52)(27,53)(33,66)(34,62)(35,65)(36,63)(37,64)(38,60)(40,67)(41,58)(44,68)(45,59)(46,69)(47,61)(49,50)(55,72)(56,71)(57,70)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cfg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32761);