#6574249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,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, 6481445);