#7143816

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7051012);