#6533404

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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3311 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)(13,14,19,23)(15,24,27,34)(16,21,28,36)(17,22,18,31)(20,32,35,26)(25,38,39,44)(29,37,30,41)(33,42,43,40);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,23,35)(16,25,28,39)(17,26,18,32)(21,33,36,43)(22,34,31,24)(29,40,30,42)(37,44,41,38);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,28)(20,33)(22,37)(23,36)(24,38)(26,40)(27,39)(31,41)(32,42)(34,44)(35,43);(5,7)(6,10)(8,11)(9,12)(16,28)(21,36)(25,39)(29,30)(33,43)(37,41)(38,44)(40,42);(7,11)(10,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,28,30)(20,34,35,24)(21,37,36,41)(25,40,39,42)(33,44,43,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6440600);