#6866117

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(19,31)(20,27)(21,37)(22,30)(23,29)(24,28)(25,35)(26,36)(32,46)(33,41)(34,42)(38,43)(39,44)(40,47)(45,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,25)(20,32)(21,33)(22,39)(23,35)(24,34)(26,40)(27,41)(28,44)(29,31)(30,42)(36,48)(37,46)(38,47)(43,45);(3,8)(5,7)(10,14)(15,16)(17,20,23,21)(18,26,29,43)(19,32,35,33)(22,36,24,38)(25,40,31,45)(27,30,37,28)(34,47,39,48)(41,42,46,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,27,29,37)(19,33,35,32)(22,36,24,38)(25,41,31,46)(26,30,43,28)(34,47,39,48)(40,42,45,44);(9,12,13,11)(10,15,14,16)(17,22,23,24)(18,28,29,30)(19,34,35,39)(20,36,21,38)(25,42,31,44)(26,27,43,37)(32,47,33,48)(40,46,45,41)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6773313);