Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23573 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18)(19,27)(20,24)(21,30)(22,25)(23,28)(26,31)(29,32);(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,22)(21,29)(24,26)(25,32)(27,30)(28,31);(2,5)(6,8)(10,13)(14,15)(17,20,22,19)(18,24,25,27)(21,29,31,28)(23,30,32,26);(9,11,12,16)(10,14,13,15)(19,20)(23,32)(24,27)(28,29);(1,3,4,7)(2,6,5,8)(18,25)(23,32)(24,27)(26,30)Polycyclic presentation:
2^9|c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281810, #72308665, #72308900, … |
| maximal normal subgroup of | #72281810, #72308665, #72308900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371558);