Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14415 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,3,12,10)(4,13,18,26,17,8,16,24)(5,19,21,9,22,20,25,14)(15,28,31,27,30,32,29,23);(1,3)(2,7)(4,8,17,13)(5,20)(6,10)(9,25)(11,12)(14,21)(15,28,30,32)(16,26,18,24)(19,22)(23,31,27,29);(1,4,6,18,7,17,12,16)(2,8,10,26,3,13,11,24)(5,15,21,31,22,30,25,29)(9,23,19,32,14,27,20,28);(1,5,6,21,7,22,12,25)(2,9,10,19,3,14,11,20)(4,15,18,31,17,30,16,29)(8,23,26,32,13,27,24,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72339028, #72339034, #72339042 |
| maximal normal subgroup of | #72339028, #72339034, #72339042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293002);