#7149328

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1931 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)(7,11)(10,12)(13,14)(15,23)(16,27)(17,31)(18,21)(19,25)(20,29)(22,32)(24,39)(26,38)(28,37)(30,36)(33,41)(34,43)(35,42)(40,44);(6,9)(7,11)(13,15,17,25,18,26,32,43)(14,19,22,23,21,34,31,38)(16,28,30,41,29,24,40,42)(20,35,36,39,27,33,44,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,24)(17,30)(18,29)(19,33)(21,27)(22,36)(23,37)(25,42)(26,28)(31,44)(32,40)(34,35)(38,39)(41,43);(6,9)(10,12)(15,25)(16,29)(17,32)(19,23)(20,27)(22,31)(24,41)(26,43)(28,42)(33,39)(34,38)(35,37);(5,7,8,11)(6,10,9,12)(14,21)(15,26)(16,29)(22,31)(25,43)(30,40)(33,35)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 7056524);