#7002797

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(17,18,19,27,22,26,30,23)(20,31,32,40,34,38,28,24)(21,25,29,43,36,42,46,37)(33,47,45,41,35,48,44,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,46)(24,38,40,31)(25,37,42,43)(33,45,35,44)(39,48,41,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,34)(18,24,26,40)(19,28,30,32)(21,35,36,33)(23,31,27,38)(25,41,42,39)(29,45,46,44)(37,48,43,47);(3,8)(5,7)(11,16)(13,15)(17,21,22,36)(18,25,26,42)(19,29,30,46)(20,33,34,35)(23,37,27,43)(24,39,40,41)(28,44,32,45)(31,47,38,48);(10,14)(13,15)(18,26)(21,36)(23,27)(24,40)(29,46)(31,38)(33,35)(44,45)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6909993);