#6350581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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

2959 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)(10,11)(12,20)(13,14)(16,23)(17,18)(19,24);(1,2)(3,6)(4,8)(5,7)(9,10,15,18)(11,13,17,14)(12,16,20,24)(19,21,23,22);(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24);(1,3,5,4)(2,6,7,8)(9,12,15,20)(10,16,18,24)(11,19,17,23)(13,21,14,22);(12,20)(16,24)(19,23)(21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257777);