#6749329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,31)(21,35)(22,27)(23,24)(25,29)(26,37)(30,38)(32,41)(33,46)(34,39)(36,42)(40,45)(43,48)(44,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,36,37,30)(25,39,41,31)(26,42,35,38)(33,43,47,45)(40,44,48,46);(11,15)(13,16)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,33,37,47)(24,31,28,39)(26,40,35,48)(30,43,36,45)(38,44,42,46);(1,3,5,4)(2,6,7,8)(17,21,22,37)(18,26,27,35)(19,30,23,36)(20,33,34,47)(24,38,28,42)(25,40,41,48)(29,43,32,45)(31,44,39,46);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(18,27)(21,37)(24,28)(25,41)(30,36)(31,39)(33,47)(43,45)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6656525);