#694461

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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
prime radical	C₂²×#538	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4346 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 601657);