#1782140

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1988 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)(3,6)(4,7)(5,8)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21)(27,31)(28,32);(1,3)(2,5)(4,8)(6,7)(10,12)(14,15)(17,18,20,22)(19,21,24,23)(25,26,29,30)(27,32,31,28);(3,8)(5,6)(9,10,13,12)(11,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,27)(26,28)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,16)(10,14,12,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,30)(31,32);(1,2)(3,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,28)(26,27)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fghi,d^2=fghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1689336);