#6592849

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1695 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,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, 6500045);