#6598407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1831 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505603);