#6349011

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348967
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256207);