#3407348

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1618 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgi,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

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, 3314544);