#7055077

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,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, 6962273);