Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395191 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14722 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1800 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)(3,14)(4,9)(5,13)(6,11)(7,12)(8,16)(10,18)(15,29)(17,23)(19,27)(20,26)(21,25)(22,32)(24,30)(28,31);(1,3,5,16,7,17,6,18)(2,8,11,14,12,10,13,23)(4,15,19,30,20,31,21,32)(9,24,25,29,26,22,27,28);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,22)(10,24)(11,25)(12,26)(13,27)(14,28)(16,30)(17,31)(18,32)(23,29);(1,3,5,16,7,17,6,18)(2,10,11,23,12,8,13,14)(4,15,19,30,20,31,21,32)(9,24,25,29,26,22,27,28)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216192, #72216320, … |
| maximal normal subgroup of | #72215972, #72216192, #72216320, … |
| maximal quotient of | #72237186, C2×#395191, #72252135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302387);