Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14201 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1110 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)(4,18,21,27)(5,10)(6,11)(7,12)(8,15)(9,20,28,22)(14,23,30,32)(16,25)(17,24)(19,26,29,31);(1,3,5,15,7,17,6,16)(2,8,11,13,12,25,10,24)(4,19,20,14,21,29,22,30)(9,26,27,32,28,31,18,23);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,23)(10,18)(11,27)(12,28)(13,26)(15,29)(16,19)(17,30)(24,31)(25,32);(2,10,12,11)(8,24,25,13)(9,18,28,27)(23,31,32,26)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=efh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213889, #72219485, #72219493 |
| maximal normal subgroup of | #72213889, #72219485, #72219493 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284968);