#2805612

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=fgh,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,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, 2712808);