#1259526

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

2744 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)(11,16)(12,14)(17,18,22,23)(19,26,24,20)(21,29,31,25)(27,28,32,30);(5,6,7,8)(9,10,13,15)(11,14,16,12)(18,23)(19,24)(25,29)(27,32);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,24)(18,20,23,26)(21,27,31,32)(25,28,29,30);(1,2)(3,4)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32);(9,12,13,14)(10,11,15,16)(17,21,22,31)(18,25,23,29)(19,27,24,32)(20,28,26,30)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,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, 1166722);