#1287256

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2140 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=gh,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, 1194452);