#448253

order	512
nilpotency class	3
isoclinism	isoclinic class of #419459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=e,d^2=eg,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355449);