Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2407 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)(3,8)(4,16)(5,7)(6,11)(9,18)(10,20)(12,24)(13,23)(14,21)(15,25)(17,22)(19,26)(27,32)(28,30)(29,31);(2,7)(3,12)(4,17)(5,20)(9,16)(10,11)(13,28)(14,15)(18,19)(21,24)(23,31)(30,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,21)(9,23)(10,24)(11,25)(12,20)(16,30)(17,27)(18,28)(19,29)(22,31)(26,32);(1,4)(2,9)(3,13)(5,19)(6,18)(7,22)(8,23)(10,26)(11,16)(12,27)(14,29)(15,28)(17,20)(21,31)(24,32)(25,30)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240440, #72240441, … |
| maximal normal subgroup of | #72237642, #72240440, #72240441, … |
| maximal quotient of | #72237648, C2×#417667, #72251245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324863);