#6866134

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(17,18,22,24)(19,29,30,42)(20,33,34,26)(21,28,23,27)(25,37,38,32)(31,45,44,43)(35,40,36,41)(39,47,48,46);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(18,24)(19,30)(20,34)(27,28)(32,37)(35,36)(39,48)(43,45);(3,8)(5,7)(10,14)(15,16)(17,19,22,30)(18,25,24,38)(20,31,34,44)(21,32,23,37)(26,39,33,48)(27,29,28,42)(35,46,36,47)(40,45,41,43);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,36)(22,34)(23,35)(24,33)(25,39)(27,41)(28,40)(29,43)(30,44)(32,47)(37,46)(38,48)(42,45);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,24,28)(19,32,30,37)(20,35,34,36)(25,29,38,42)(26,40,33,41)(31,46,44,47)(39,45,48,43)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6773330);