#124462

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,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, 31658);