#7054885

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2115 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)(3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(17,30)(20,33)(21,26)(22,31)(25,41)(28,32)(34,39)(35,42)(36,44)(37,38)(40,43);(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,20,24,35)(16,26,29,38)(17,31,30,19)(21,28,37,32)(23,39,27,44)(25,42,41,33)(34,40,36,43);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,27)(17,25)(18,24)(19,33)(21,36)(22,35)(23,29)(26,44)(28,43)(30,41)(31,42)(32,40)(34,37)(38,39);(7,12)(9,11)(13,16,18,29)(14,21,22,37)(15,23,24,27)(17,32,30,28)(19,26,31,38)(20,34,35,36)(25,43,41,40)(33,39,42,44);(6,10)(9,11)(14,22)(15,24)(19,31)(21,37)(23,27)(25,41)(26,38)(40,43)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

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, 6962081);