#6601424

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(17,18,22,26)(19,23,28,38)(20,30,32,42)(21,34,36,25)(24,35,41,33)(27,43,31,48)(29,46,45,39)(37,47,40,44);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,28)(18,23,26,38)(20,31,32,27)(21,29,36,45)(24,40,41,37)(25,39,34,46)(30,48,42,43)(33,47,35,44);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20,22,32)(18,24,26,41)(19,27,28,31)(21,35,36,33)(23,37,38,40)(25,42,34,30)(29,47,45,44)(39,48,46,43);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,28)(20,32)(23,38)(24,41)(29,45)(30,42)(33,35)(39,46);(10,13)(11,15)(19,28)(20,32)(23,38)(24,41)(29,45)(30,42)(33,35)(39,46)

Polycyclic presentation:

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

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, 6508620);