#134392

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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	#1804
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

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, 41588);