#6606082

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

2080 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)(16,27,30,41)(17,22,18,32)(21,28,37,29)(24,38,40,43)(25,34,31,36)(33,42,44,39);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,24)(17,31)(18,25)(19,26)(21,33)(22,36)(23,35)(27,38)(28,42)(29,39)(30,40)(32,34)(37,44)(41,43);(6,9)(7,11)(13,16,17,28,19,30,18,29)(14,21,32,41,23,37,22,27)(15,24,31,42,26,40,25,39)(20,33,34,43,35,44,36,38);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,28,30,29)(20,34,35,36)(21,27,37,41)(24,39,40,42)(33,43,44,38);(6,9)(7,11)(15,26)(20,35)(24,40)(25,31)(33,44)(34,36)(38,43)(39,42)

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,d^b=di,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, 6513278);