#5025900

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

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, 4933096);