#6533241

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×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,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, 6440437);