#6795998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1328 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)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18,23,29,24,30,22,28)(19,25,32,41,33,42,38,45)(20,27,35,44,21,36,37,26)(31,46,48,43,34,40,47,39);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19,24,33)(18,25,30,42)(20,34,21,31)(22,38,23,32)(26,43,44,39)(27,40,36,46)(28,45,29,41)(35,47,37,48);(10,13)(14,16)(17,20,24,21)(18,26,30,44)(19,31,33,34)(22,35,23,37)(25,39,42,43)(27,29,36,28)(32,47,38,48)(40,45,46,41);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,30,36)(19,31,33,34)(22,35,23,37)(25,40,42,46)(26,29,44,28)(32,48,38,47)(39,45,43,41);(1,5,6,4)(2,7,3,8)(17,22,24,23)(18,28,30,29)(19,32,33,38)(20,35,21,37)(25,41,42,45)(26,27,44,36)(31,47,34,48)(39,40,43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6703194);