#7143828

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

2221 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)(13,14,18,22)(15,23,27,39)(16,20,29,36)(17,31,24,21)(19,32,34,26)(25,38,28,43)(30,44,40,37)(33,41,35,42);(6,9)(7,11)(13,15,24,32,18,27,17,26)(14,19,31,39,22,34,21,23)(16,28,40,42,29,25,30,41)(20,35,44,38,36,33,37,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,29)(14,20,22,36)(15,25,27,28)(17,30,24,40)(19,33,34,35)(21,37,31,44)(23,38,39,43)(26,42,32,41);(6,9)(10,12)(15,26)(16,29)(17,24)(19,23)(20,36)(21,31)(25,41)(27,32)(28,42)(33,43)(34,39)(35,38);(5,7,8,11)(6,10,9,12)(15,27)(16,29)(19,34)(20,36)(23,39)(26,32)(30,40)(37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,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, 7051024);