#6533530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

3281 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)(13,14,19,24)(15,25,28,34)(16,21,30,38)(17,23,18,22)(20,32,35,27)(26,39,42,43)(29,37,31,36)(33,40,44,41);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,25)(23,34)(24,35)(29,40)(30,42)(31,41)(36,43)(37,39)(38,44);(2,4)(6,9)(10,12)(13,16,19,30)(14,21,24,38)(15,26,28,42)(17,31,18,29)(20,33,35,44)(22,37,23,36)(25,39,34,43)(27,41,32,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,30,31)(20,25,35,34)(21,36,38,37)(26,40,42,41)(33,43,44,39);(7,11)(10,12)(13,17,19,18)(14,23,24,22)(15,27,28,32)(16,29,30,31)(20,34,35,25)(21,37,38,36)(26,40,42,41)(33,39,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440726);