#6628067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,12,16)(17,18)(19,24)(20,31)(21,27)(22,35)(23,26)(25,32)(28,43)(29,39)(30,38)(33,42)(34,40)(36,41)(37,45)(44,47)(46,48);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,23,30)(18,24,26,38)(20,28,33,46)(21,34,22,29)(25,37,41,47)(27,40,35,39)(31,43,42,48)(32,45,36,44);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,46)(31,35)(34,45)(38,47)(39,43)(40,48);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(18,26)(24,38)(25,41)(27,35)(31,42)(37,47)(39,40)(43,48);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,27,26,35)(19,29,30,34)(20,32,33,36)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6535263);