Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421918 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15299 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #23444 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1351 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)(3,7,13,21)(4,8,15,23)(5,16,18,9)(6,10)(11,19)(12,20)(14,22)(17,28)(24,32,26,30)(25,31)(27,29);(1,3)(2,7,10,21)(4,11,15,25)(5,17,18,12)(6,13)(8,19)(9,20)(14,26)(16,28)(22,30,29,32)(23,31)(24,27);(1,4)(2,8,10,23)(3,11,13,25)(5,14)(6,15)(7,19)(9,22,16,29)(12,24,17,26)(18,27)(20,30)(21,31)(28,32);(1,5)(2,9)(3,12)(4,14)(6,18)(7,20)(8,22)(10,16)(11,24)(13,17)(15,27)(19,30)(21,28)(23,29)(25,26)(31,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237732, #72237762, #72237763, … |
| maximal normal subgroup of | #72237732, #72237762, #72237763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329114);