#6850752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1423 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6757948);