#1411978

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
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

1562 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,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, 1319174);