#6771768

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1569 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)(17,18,22,26)(19,27,28,40)(20,24,31,38)(21,33,35,25)(23,34,36,30)(29,41,42,48)(32,45,44,39)(37,46,47,43);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(19,28)(23,36)(27,40)(29,42)(30,34)(37,47)(41,48)(43,46);(11,16)(13,15)(17,19,22,28)(18,23,26,36)(20,29,31,42)(21,34,35,30)(24,37,38,47)(25,40,33,27)(32,46,44,43)(39,48,45,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,22,31)(18,24,26,38)(19,29,28,42)(21,32,35,44)(23,37,36,47)(25,39,33,45)(27,41,40,48)(30,43,34,46);(2,5)(6,8)(20,31)(24,38)(29,42)(32,44)(37,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

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

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, 6678964);