#1297472

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(6,8)(17,18,22,21)(19,25,27,23)(20,24,29,28)(26,31,32,30);(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31);(5,6,7,8)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32);(1,2)(3,4)(9,11,12,15)(10,13,14,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=i,e^2=fh,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, 1204668);