#6357321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357317
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2271 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)(3,7)(6,8)(10,11)(13,14)(16,19)(17,18)(21,22)(23,24);(2,6)(5,8)(9,10,13,11,15,18,14,17)(12,16,21,19,20,23,22,24);(1,2,3,5,4,6,7,8)(9,11,13,18,15,17,14,10)(12,19,21,23,20,24,22,16);(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=f,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ci,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264517);