#1126167

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1932 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fh,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

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, 1033363);