Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #378118 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14207 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #23444 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1576 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)(4,15,18,8)(5,19)(6,10)(7,14)(9,20)(12,23,16,21)(13,25)(17,28,30,24)(22,27)(26,31,29,32);(1,3)(2,7,10,11)(4,16,18,12)(5,13)(6,14)(8,23)(9,22,19,25)(15,21)(17,29,30,26)(20,27)(24,32)(28,31);(1,4)(2,8)(3,12)(5,17)(6,18)(7,21)(9,24)(10,15)(11,23)(13,26)(14,16)(19,28)(20,30)(22,31)(25,32)(27,29);(1,5)(2,9,10,19)(3,13)(4,17)(6,20)(7,22,11,25)(8,24,15,28)(12,26)(14,27)(16,29)(18,30)(21,31,23,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=efh,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237787, #72237788, #72237790, … |
| maximal normal subgroup of | #72237787, #72237788, #72237790, … |
| maximal quotient of | #72265864 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285314);