#6866183

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

1409 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)(20,32)(21,27)(22,24)(23,36)(25,30)(26,34)(29,44)(31,40)(33,42)(35,41)(37,38)(39,45)(43,46)(47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(18,24)(25,38)(26,32)(27,36)(28,40)(39,48)(41,42)(43,44);(3,8)(5,7)(10,14)(15,16)(17,19,22,31)(18,25,24,38)(20,29,34,46)(21,30,23,37)(26,39,32,48)(27,40,36,28)(33,45,35,47)(41,43,42,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,29)(21,35)(22,34)(23,33)(24,32)(25,39)(27,42)(28,43)(30,47)(31,46)(36,41)(37,45)(38,48)(40,44);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,24,36)(19,30,31,37)(20,33,34,35)(25,40,38,28)(26,41,32,42)(29,45,46,47)(39,43,48,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fhi,e^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, 6773379);