#6362756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362750
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

1786 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)(10,15)(11,18)(12,13)(16,19)(17,21)(23,24);(9,10,14,17)(11,19,18,23)(12,21,13,15)(16,22,24,20);(2,5)(3,8)(6,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,24)(19,20)(22,23);(9,11,14,18)(10,16,17,24)(12,22,13,20)(15,23,21,19);(2,6)(5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269952);