#6534128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1719 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)(3,6)(4,5)(7,8)(17,18,23,27)(19,24,29,38)(20,31,32,42)(21,26,22,36)(25,33,41,34)(28,43,45,47)(30,39,35,40)(37,46,48,44);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,38)(20,28,32,45)(21,35,22,30)(25,37,41,48)(26,40,36,39)(31,43,42,47)(33,46,34,44);(10,13)(14,16)(17,20,23,32)(18,25,27,41)(19,28,29,45)(21,34,22,33)(24,37,38,48)(26,42,36,31)(30,46,35,44)(39,43,40,47);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,29)(20,32)(24,38)(25,41)(30,35)(31,42)(33,34)(39,40);(11,15)(14,16)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,33,32,34)(24,39,38,40)(25,31,41,42)(28,44,45,46)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6441324);