#2516510

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

1913 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=g,d^2=g,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, 2423706);