#1402811

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

1450 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)(17,18,21,24)(19,23,20,22)(25,26,29,30)(27,31,35,33)(28,36,38,32)(34,40,37,39);(1,2,6,4)(3,7,5,8)(10,13)(14,16)(26,30)(31,33)(32,36)(39,40);(1,3,6,5)(2,7,4,8)(9,10,12,13)(11,14,15,16)(17,19,21,20)(18,22,24,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,4,6,2)(3,8,5,7)(17,19,21,20)(18,22,24,23)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,23,24,22)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=gh,d^2=g,e^2=fgh,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, 1310007);