#6353504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353473
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3203 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:

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

Polycyclic presentation:

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

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, 6260700);