#1899642

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1434 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fghi,c^2=gi,d^2=fghi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1806838);