#7143810

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

2279 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)(14,19)(17,20)(21,33)(22,28)(23,34)(24,40)(27,35)(30,32)(36,44)(37,41)(38,39)(42,43);(6,9)(7,11)(13,14,20,34,18,23,17,19)(15,24,35,33,26,21,27,40)(16,28,32,38,31,39,30,22)(25,41,43,36,29,44,42,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,29)(17,27)(18,26)(19,33)(20,35)(22,37)(23,24)(25,31)(28,41)(30,43)(32,42)(34,40)(36,38)(39,44);(6,9)(10,12)(13,16,18,31)(14,22,23,38)(15,25,26,29)(17,32,20,30)(19,28,34,39)(21,36,24,37)(27,43,35,42)(33,44,40,41);(5,7,8,11)(6,10,9,12)(14,23)(15,26)(19,34)(22,38)(25,29)(27,35)(28,39)(42,43)

Polycyclic presentation:

2^9|b^2=fi,d^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, 7051006);