#2433673

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1839 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fhi,d^2=fhi,e^2=fgh,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, 2340869);