#6350429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4010 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:

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

Polycyclic presentation:

2^9|b^2=i,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, 6257625);