Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392727 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,12)(3,13,17,23,16,26,15,8)(4,18,20,24,21,9,19,25)(14,27,30,22,29,32,28,31);(1,3)(2,8,11,23)(4,14)(5,15)(6,17)(7,16)(9,22,18,31)(10,13,12,26)(19,28)(20,30)(21,29)(24,27,25,32);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,24)(11,18)(12,25)(13,27)(15,30)(16,29)(17,28)(23,31)(26,32);(2,10)(5,6)(8,13)(9,24)(11,12)(15,17)(18,25)(19,20)(22,27)(23,26)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233775, #72237692, … |
| maximal normal subgroup of | #72233772, #72233775, #72237692, … |
| maximal quotient of | #72237702, #72249701, C2×#392729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299925);