#6745795

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6652991);