#1293520

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1200716);