#6573450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,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, 6480646);