Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15528 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
899 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)(3,11,14,7)(4,15)(5,19,20,9)(6,10)(8,18)(12,21,16,25)(13,24)(17,23,30,28)(22,27)(26,31)(29,32);(1,3)(2,7)(4,16,18,12)(5,13)(6,14)(8,21,15,25)(9,24)(10,11)(17,26,30,29)(19,22)(20,27)(23,31,28,32);(1,4)(2,8,10,15)(3,12)(5,17)(6,18)(7,21,11,25)(9,23,19,28)(13,26)(14,16)(20,30)(22,31,24,32)(27,29);(1,5)(2,9)(3,13)(4,17)(6,20)(7,22)(8,23)(10,19)(11,24)(12,26)(14,27)(15,28)(16,29)(18,30)(21,31)(25,32)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237754, #72237762, #72237765, … |
| maximal normal subgroup of | #72237754, #72237762, #72237765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342718);