#6606070

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2076 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)(15,20,26,35)(16,28,31,42)(17,22,18,32)(21,29,36,30)(24,37,39,44)(25,34,27,41)(33,38,43,40);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,33)(22,34)(23,35)(28,37)(29,38)(30,40)(31,39)(32,41)(36,43)(42,44);(6,9)(7,11)(13,16,17,29,19,31,18,30)(14,21,32,42,23,36,22,28)(15,24,25,38,26,39,27,40)(20,33,41,44,35,43,34,37);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,25,26,27)(16,29,31,30)(20,34,35,41)(21,28,36,42)(24,38,39,40)(33,37,43,44);(6,9)(7,11)(15,26)(20,35)(24,39)(25,27)(33,43)(34,41)(37,44)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513266);