#6994217

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

2228 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)(5,6)(7,9)(8,10)(11,12)(14,19)(17,29)(20,32)(21,26)(22,30)(25,40)(27,31)(33,37)(34,41)(35,36)(39,42)(43,44);(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,20,24,34)(16,26,28,35)(17,30,29,19)(21,27,36,31)(23,37,38,44)(25,41,40,32)(33,39,43,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,20,22,34)(16,23,28,38)(17,25,29,40)(19,32,30,41)(21,33,36,43)(26,37,35,44)(27,39,31,42);(7,12)(9,11)(13,16,18,28)(14,21,22,36)(15,23,24,38)(17,31,29,27)(19,26,30,35)(20,33,34,43)(25,42,40,39)(32,37,41,44);(6,10)(9,11)(15,24)(20,34)(23,38)(25,40)(32,41)(33,43)(37,44)(39,42)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,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, 6901413);