Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456638 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16284 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10)(3,6,11,23)(4,14,17,8)(5,13)(9,22)(12,16,25,18)(15,30)(19,29)(20,28)(21,27)(24,32)(26,31);(1,3,6,10,7,11,23,2)(4,12,16,8,17,25,18,14)(5,13,22,28,21,27,20,9)(15,26,32,24,31,30,19,29);(1,4)(2,8)(3,12)(5,19)(6,18)(7,17)(9,26)(10,14)(11,25)(13,29)(15,20)(16,23)(21,32)(22,31)(24,27)(28,30);(1,5)(2,9)(3,13)(4,15)(6,22)(7,21)(8,24)(10,28)(11,27)(12,26)(14,29)(16,32)(17,31)(18,19)(20,23)(25,30)Polycyclic presentation:
2^9|a^2=f,b^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237069, #72237073, #72237074, … |
| maximal normal subgroup of | #72237069, #72237073, #72237074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363835);