#6454050

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437621
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24);(9,11)(10,14)(12,17)(13,19)(15,22)(16,23)(18,20)(21,24);(2,5)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23);(9,12)(10,15)(11,18)(13,20)(14,22)(16,24)(17,19)(21,23);(1,3,4,7)(2,6,5,8)(11,19)(14,23)(17,18)(21,22)

Polycyclic presentation:

2^9|e^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^c=fi,f^d=fi

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, 6361246);