#6660861

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660434
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	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6568057);