#135734

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1846
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

236 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42930);