Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454093 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16249 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2227 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)(3,12)(4,9)(5,21)(6,10)(7,11)(8,14)(13,24)(15,28)(16,23)(17,32)(18,26)(19,25)(20,27)(22,30)(29,31);(1,3)(2,8)(4,17)(5,16)(6,14)(7,15)(9,24)(10,28)(11,23)(12,21)(13,18)(19,31)(20,30)(22,26)(25,32)(27,29);(1,4)(2,9)(3,13)(5,18)(6,19)(7,20)(8,22)(10,25)(11,27)(12,29)(14,17)(15,31)(16,30)(21,26)(23,32)(24,28);(4,18)(5,6)(9,25)(10,21)(12,28)(13,17)(14,16)(19,20)(22,29)(24,32)(26,27)(30,31)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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216142, #72216154, … |
| maximal normal subgroup of | #72215995, #72216142, #72216154, … |
| maximal quotient of | C2×#454093, #72252105, #72317314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361289);