#6772049

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,33)(28,43)(29,38)(30,39)(32,41)(34,42)(35,40)(37,45)(44,48)(46,47);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,23,29)(18,24,27,38)(20,28,34,44)(21,35,22,30)(25,37,41,47)(26,40,36,39)(31,43,42,48)(32,46,33,45);(11,16)(13,15)(17,20,23,34)(18,25,27,41)(19,28,29,44)(21,33,22,32)(24,37,38,47)(26,42,36,31)(30,46,35,45)(39,43,40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,29)(24,38)(28,44)(30,35)(37,47)(39,40)(43,48)(45,46);(2,5)(6,8)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,32,34,33)(24,39,38,40)(25,31,41,42)(28,45,44,46)(37,48,47,43)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fh,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=efhi,e^c=ei,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, 6679245);