#3325203

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
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

1486 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)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,20,23,22)(25,26)(27,33)(28,36)(29,32)(30,35)(31,38)(34,39)(37,40);(11,16)(13,15)(19,23)(20,22)(25,26,29,32)(27,30,35,33)(28,36,38,31)(34,40,37,39);(1,2,5,4)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,22,24,20)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,3,5,7)(2,6,4,8)(9,11,12,16)(10,13,14,15)(17,20,21,22)(18,23,24,19)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fgi,d^2=hi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 3232399);