#1186893

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1490 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fgh,e^2=gh,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, 1094089);