#6768347

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=hi,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=efh,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, 6675543);