#135735

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

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

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,c^b=ci,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, 42931);