#6513014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500019
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3343 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420210);