Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427241 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2471 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15369 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1295 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)(3,11,13,7)(4,15)(5,18)(6,9)(8,17)(10,19)(12,25,26,29)(14,21,28,24)(16,30)(20,27,23,32)(22,31);(7,11)(12,26)(21,24)(27,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,20)(9,11)(10,21)(15,29)(16,26)(17,27)(18,24)(19,28)(22,23)(25,30)(31,32);(1,4)(2,8,9,22)(3,12)(5,17)(6,16)(7,20,11,23)(10,15,18,30)(13,26)(14,27)(19,31)(21,29,24,25)(28,32)Polycyclic presentation:
2^9|a^2=e,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237065, #72237661, … |
| maximal normal subgroup of | #72237063, #72237065, #72237661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334577);