#6633294

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 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)(10,14)(11,12)(17,18,23,28)(19,29,32,42)(20,25,33,39)(21,27,22,26)(24,36,38,31)(30,43,44,48)(34,41,35,40)(37,45,47,46);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(20,33)(25,39)(30,44)(34,35)(37,47)(40,41)(43,48)(45,46);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,19,23,32)(18,24,28,38)(20,30,33,44)(21,31,22,36)(25,37,39,47)(26,29,27,42)(34,45,35,46)(40,48,41,43);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,33)(24,37)(26,41)(27,40)(28,39)(29,43)(31,46)(32,44)(36,45)(38,47)(42,48);(2,6)(5,8)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,48,47,43)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6540490);