Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469875 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2607 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16666 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
864 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)(3,7)(4,14,16,29)(5,10)(6,9)(8,17,24,32)(11,27,15,30)(12,20)(13,22)(18,26)(19,25)(21,28,23,31);(7,20)(21,23)(22,26)(27,30);(1,3)(2,7)(4,15,16,11)(5,18)(6,12)(8,23,24,21)(9,20)(10,26)(13,19)(14,27,29,30)(17,28,32,31)(22,25);(1,4)(2,8,9,24)(3,11)(5,17)(6,16)(7,21,20,23)(10,14,25,29)(12,15)(13,28)(18,31)(19,32)(22,27,26,30)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251676, #72251684, #72251685, … |
| maximal normal subgroup of | #72251676, #72251684, #72251685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377071);