#6775040

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(17,18,23,28)(19,24,31,40)(20,33,21,26)(22,36,37,27)(25,35,43,34)(29,44,30,39)(32,48,47,41)(38,46,42,45);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,28,40)(20,30,21,29)(22,32,37,47)(25,42,43,38)(26,44,33,39)(27,41,36,48)(34,46,35,45);(10,13)(14,16)(17,20,23,21)(18,25,28,43)(19,29,31,30)(22,35,37,34)(24,38,40,42)(26,36,33,27)(32,46,47,45)(39,48,44,41);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,26,28,33)(19,30,31,29)(22,35,37,34)(24,39,40,44)(25,36,43,27)(32,46,47,45)(38,48,42,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,31)(20,21)(24,40)(25,43)(26,33)(32,47)(34,35)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=f,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, 6682236);