#6795591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(17,18,23,36,21,26,22,27)(19,28,37,39,30,24,31,43)(20,32,35,25,33,41,34,40)(29,44,47,42,45,48,46,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,37)(25,38)(26,28)(27,39)(32,48)(33,45)(34,46)(35,47)(36,43)(40,42)(41,44);(2,5)(6,8)(10,13)(14,15)(17,20,21,33)(18,25,26,40)(19,29,30,45)(22,35,23,34)(24,38,28,42)(27,41,36,32)(31,47,37,46)(39,44,43,48);(1,3,4,7)(2,6,5,8)(18,26)(19,30)(25,40)(27,36)(29,45)(31,37)(32,41)(46,47);(9,11,12,16)(10,14,13,15)(19,30)(24,28)(29,45)(31,37)(38,42)(39,43)(44,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6702787);