#6867447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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)(20,31,29,39,33,42,36,25)(21,26,35,40,34,41,30,37)(32,44,43,47,45,48,46,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,27,28)(20,29,33,36)(21,30,34,35)(25,31,39,42)(26,37,41,40)(32,43,45,46)(38,44,47,48);(10,14)(15,16)(17,20,23,33)(18,25,27,39)(19,29,22,36)(21,32,34,45)(24,31,28,42)(26,38,41,47)(30,43,35,46)(37,44,40,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,34)(18,26,27,41)(19,30,22,35)(20,32,33,45)(24,37,28,40)(25,38,39,47)(29,43,36,46)(31,44,42,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(18,27)(21,34)(24,28)(25,39)(30,35)(31,42)(32,45)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6774643);