#2044392

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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

1452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(25,26,29,30)(27,33,34,31)(28,32,36,37)(35,39,40,38);(2,6)(4,8)(10,13)(14,16)(17,18,21,20)(19,23,24,22)(26,30)(27,34)(28,36)(38,39);(1,3,5,7)(2,4,6,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,27,29,34)(26,31,30,33)(28,35,36,40)(32,38,37,39);(1,4)(2,7)(3,6)(5,8)(25,28,29,36)(26,32,30,37)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=fh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1951588);