#6348980

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3102 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:

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

Polycyclic presentation:

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

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, 6256176);