#6800063

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 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)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,29,32,24)(20,33,36,44)(21,28,23,26)(25,38,42,35)(30,45,34,48)(31,40,37,43)(39,47,41,46);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,34)(21,37)(22,32)(23,31)(25,41)(26,43)(27,29)(28,40)(30,36)(33,45)(35,46)(38,47)(39,42)(44,48);(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,38)(24,39)(26,33)(27,42)(28,44)(29,41)(31,46)(32,34)(37,47)(40,45)(43,48);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,35,36,38)(24,40,29,43)(25,33,42,44)(30,46,34,47)(39,45,41,48);(1,3,4,7)(2,6,5,8)(18,27)(24,29)(25,42)(26,28)(33,44)(39,41)(40,43)(45,48)

Polycyclic presentation:

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

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, 6707259);