Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,14)(4,10)(5,12)(6,11)(7,13)(8,17)(9,19)(15,26)(16,25)(18,24)(20,29)(21,27)(22,30)(23,32)(28,31);(1,3)(2,8)(4,15)(5,18)(6,17)(7,16)(9,26)(10,28)(11,25)(12,14)(13,24)(19,32)(20,30)(21,31)(22,27)(23,29);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,22)(10,11)(12,29)(13,27)(14,28)(16,30)(17,31)(18,32)(23,25)(24,26);(1,4)(2,10)(3,15)(5,21)(6,19)(7,20)(8,23)(9,11)(12,27)(13,29)(14,26)(16,30)(17,32)(18,31)(22,25)(24,28);(3,16)(8,24)(14,25)(15,30)(17,18)(22,26)(23,28)(31,32)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,g^c=gi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233774, #72233775, … |
| maximal normal subgroup of | #72233772, #72233774, #72233775, … |
| maximal quotient of | #72214896, C2×#6437576, #72238124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344772);