#7064664

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,24,19,25,17,23)(15,20,28,37,29,38,33,41)(16,30,31,39,32,22,34,21)(26,36,43,44,27,40,42,35);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,25,38)(16,26,32,27)(17,33,18,28)(21,35,39,44)(22,40,30,36)(23,41,24,37)(31,43,34,42);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,26)(17,31)(18,34)(19,32)(20,35)(22,24)(23,30)(25,39)(27,29)(28,42)(33,43)(36,41)(37,40)(38,44);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,27)(17,34)(18,31)(19,32)(20,36)(21,24)(23,39)(25,30)(26,29)(28,42)(33,43)(35,41)(37,44)(38,40);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,28,29,33)(16,31,32,34)(20,37,38,41)(21,30,39,22)(26,42,27,43)(35,40,44,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971860);