#129507

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂×#575	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 140 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)(11,21)(12,26,28,17)(13,30)(14,36)(15,19)(16,23)(18,34)(20,33)(22,47,50,68)(24,48)(25,54)(27,57,32,65)(29,42,59,56)(31,46)(35,62)(37,43)(38,49,55,52)(39,64)(40,53)(41,60,44,66)(45,67)(51,70,61,72)(58,69,63,71);(9,11)(10,16)(12,22)(13,31)(14,37)(15,24)(17,38)(18,43)(19,40)(20,46)(21,30)(23,34)(25,33)(26,55)(27,58)(28,50)(29,61)(32,63)(35,64)(36,39)(41,70)(42,71)(44,72)(45,54)(47,57)(48,62)(49,60)(51,59)(52,66)(53,67)(56,69)(65,68);(9,12)(10,17)(11,22)(13,32)(14,29)(15,28)(16,38)(18,44)(19,26)(20,42)(21,47)(23,52)(24,50)(25,51)(27,35)(30,57)(31,63)(33,59)(34,66)(36,56)(37,61)(39,69)(40,55)(41,45)(43,72)(46,71)(48,68)(49,53)(54,70)(58,64)(60,67)(62,65)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36703);