#6763201

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=f,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,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6670397);