#156077

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63273);