#6769392

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6676588);