#6356131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1853 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:

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

Polycyclic presentation:

2^9|b^2=fh,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,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, 6263327);