#3553098

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1891 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)(3,7)(4,5)(6,8)(10,11)(12,16)(17,18,20,22)(19,23,24,21)(27,31)(28,32);(1,2,5,4)(3,6,8,7)(9,10,13,11)(12,14,16,15)(17,18,20,22)(19,21,24,23)(25,26,29,30)(27,32,31,28);(2,4)(6,7)(9,10,13,11)(12,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,27)(26,28)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,30)(31,32);(1,4,5,2)(3,6,8,7)(9,12,13,16)(10,14,11,15)(17,19)(18,21)(20,24)(22,23)(25,28)(26,27)(29,32)(30,31)

Polycyclic presentation:

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

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, 3460294);