#6362954

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362950
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1861 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)(4,6)(7,8)(10,14)(12,20)(15,21)(16,17)(18,24)(22,23);(9,10)(11,15)(12,16)(13,17)(14,20)(18,22)(19,23)(21,24);(2,5)(3,8)(6,7)(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22);(10,16)(12,20)(14,17)(15,22)(18,24)(21,23);(2,6)(5,7)(10,17)(14,16)(15,23)(21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6270150);