Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15467 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,13,20)(4,8)(5,9)(6,10)(11,21,26,18)(12,27,16,19)(14,28)(15,23)(17,24)(22,30)(25,32,29,31);(1,3)(2,7,10,20)(4,11)(5,16,17,12)(6,13)(8,21,23,18)(9,19)(14,29,30,25)(15,26)(22,32)(24,27)(28,31);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,22)(10,23)(12,25)(13,26)(16,29)(17,30)(19,31)(20,21)(24,28)(27,32);(1,5)(2,9)(3,12)(4,14)(6,17)(7,19)(8,22)(10,24)(11,25)(13,16)(15,30)(18,31)(20,27)(21,32)(23,28)(26,29)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72210026, #72210037, #72237730, … |
| maximal normal subgroup of | #72210026, #72210037, #72237730, … |
| maximal quotient of | #72228238, #72228242, #72240508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340035);