#6874007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1482 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)(3,7)(4,6)(5,8)(17,18,22,28,23,27,19,24)(20,31,29,40,33,42,37,25)(21,34,36,38,35,26,30,41)(32,45,43,39,46,44,48,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,27,28)(20,29,33,37)(21,30,35,36)(25,31,40,42)(26,38,34,41)(32,43,46,48)(39,44,47,45);(3,8)(5,7)(10,14)(15,16)(17,20,23,33)(18,25,27,40)(19,29,22,37)(21,32,35,46)(24,31,28,42)(26,39,34,47)(30,43,36,48)(38,44,41,45);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,35)(18,26,27,34)(19,30,22,36)(20,32,33,46)(24,38,28,41)(25,39,40,47)(29,43,37,48)(31,44,42,45);(9,12,13,11)(10,15,14,16)(18,27)(21,35)(24,28)(25,40)(30,36)(31,42)(32,46)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,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, 6781203);