#6350433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
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

2942 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)(10,15)(11,17)(13,21)(16,22)(18,23)(20,24);(1,2)(3,6)(4,8)(5,7)(9,10,14,11)(12,16,19,18)(13,17,21,15)(20,23,24,22);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,18,19,16)(13,15,21,17)(20,22,24,23);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24);(12,19)(16,18)(20,24)(22,23)

Polycyclic presentation:

2^9|b^2=i,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,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

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257629);