#2490757

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	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2269 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=fg,e^2=ghi,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, 2397953);