#6654026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
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	#37379	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1585 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:

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

Polycyclic presentation:

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

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, 6561222);