#2355482

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=f,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2262678);