#6606187

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

Polycyclic presentation:

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

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, 6513383);