#1258781

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,e^2=g,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, 1165977);