#6741502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,29,36)(28,35,43,47)(32,39,44,48)(41,45,42,46);(1,3)(2,5)(4,8)(6,7)(17,19,22,30)(18,23,26,37)(20,28,31,43)(21,27,33,29)(24,35,38,47)(25,34,40,36)(32,41,44,42)(39,45,48,46);(2,6)(5,7)(10,14)(15,16)(19,27)(20,31)(21,33)(23,34)(24,38)(25,40)(28,42)(29,30)(35,46)(36,37)(41,43)(45,47);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,22,31)(18,24,26,38)(19,28,30,43)(21,32,33,44)(23,35,37,47)(25,39,40,48)(27,41,29,42)(34,45,36,46);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,27)(21,33)(23,34)(25,40)(28,41)(29,30)(32,44)(35,45)(36,37)(39,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648698);