Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423649 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15315 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1578 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)(3,13)(4,7)(5,19)(6,11)(8,12)(9,15)(10,23)(14,27)(16,17)(18,21)(20,25)(22,28)(24,26)(29,30)(31,32);(1,3)(2,9)(4,17)(5,20)(6,24)(7,15)(8,16)(10,27)(11,13)(12,26)(14,21)(18,25)(19,30)(22,29)(23,31)(28,32);(1,4,6,12,8,11,7,2)(3,9,24,17,16,26,15,13)(5,18,21,19,23,28,22,10)(14,25,31,30,29,32,20,27);(1,5,6,21,8,23,7,22)(2,10,4,18,12,19,11,28)(3,14,15,20,16,29,24,31)(9,25,13,27,26,32,17,30)Polycyclic presentation:
2^9|c^2=fi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219485, #72219493, … |
| maximal normal subgroup of | #72213913, #72219485, #72219493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330846);