#6542910

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6450106);