#7054195

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2133 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6961391);