#7003317

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

1917 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,29)(17,22)(18,34)(19,23)(20,28)(21,31)(24,26)(27,40)(30,38)(32,39)(33,36)(35,44)(37,42)(41,43);(5,6)(7,11)(8,10)(9,12)(13,15,26,33,19,17,18,28)(14,20,34,22,23,36,24,25)(16,30,41,42,31,27,32,44)(21,37,43,38,29,35,39,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,31)(14,21,23,29)(15,27,17,30)(18,32,26,41)(20,35,36,37)(22,40,25,38)(24,39,34,43)(28,42,33,44);(7,12)(9,11)(13,17,19,15)(14,22,23,25)(16,27,31,30)(18,28,26,33)(20,24,36,34)(21,38,29,40)(32,44,41,42)(35,43,37,39);(6,10)(9,11)(14,23)(16,31)(20,36)(22,25)(24,34)(27,30)(32,41)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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=df,e^b=ehi,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, 6910513);