#6966661

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 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)(17,18,21,26,22,27,37,43)(19,28,32,40,31,23,30,41)(20,33,36,42,35,25,34,24)(29,39,46,38,47,45,48,44);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,23)(20,29)(21,32)(22,31)(24,38)(25,39)(26,41)(27,28)(30,37)(33,45)(34,48)(35,47)(36,46)(40,43)(42,44);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,35)(18,24,27,42)(19,29,31,47)(21,34,37,36)(23,38,28,44)(25,43,33,26)(30,46,32,48)(39,40,45,41);(2,6)(5,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,39)(24,26)(27,33)(28,45)(30,48)(31,47)(32,46)(34,37)(38,41)(40,44)(42,43);(9,11,12,15)(10,14,13,16)(17,21,22,37)(18,26,27,43)(19,30,31,32)(20,34,35,36)(23,40,28,41)(24,25,42,33)(29,46,47,48)(38,45,44,39)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,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, 6873857);