#7054251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

2146 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)(13,14,18,22)(15,23,25,38)(16,20,28,36)(17,30,32,21)(19,31,34,26)(24,39,27,41)(29,44,43,37)(33,42,35,40);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,27)(17,31)(18,25)(20,35)(21,38)(22,34)(23,30)(24,28)(26,32)(29,40)(33,36)(37,39)(41,44)(42,43);(2,4)(7,12)(9,11)(13,16)(14,20)(15,24)(17,29)(18,28)(19,33)(21,37)(22,36)(23,39)(25,27)(26,40)(30,44)(31,42)(32,43)(34,35)(38,41);(6,10)(9,11)(15,25)(16,28)(19,34)(20,36)(23,38)(26,31)(29,43)(37,44)

Polycyclic presentation:

2^9|a^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,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961447);