#1305486

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
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

1998 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=fg,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1212682);