#6448747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3410 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,23)(13,16)(14,17)(15,21)(19,22)(20,24);(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23);(2,5)(6,8)(9,10)(11,19)(12,16)(13,23)(14,17)(15,21)(18,22)(20,24);(1,3,4,7)(2,6,5,8)(9,10)(11,19)(12,16)(13,23)(14,17)(15,21)(18,22)(20,24);(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,19,24,18)

Polycyclic presentation:

2^9|d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355943);