#7017656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1241 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 6924852);