#7043256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=gi,d^2=f,e^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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6950452);