Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,22)(16,21)(17,23)(18,19)(20,24)(25,28)(26,27);(1,2,3,4)(7,8)(10,11)(13,15,17,18)(14,19,23,22)(16,24,26,25)(20,21,28,27);(5,7,9,8)(6,10,12,11)(13,16)(14,20)(15,24)(17,26)(18,25)(19,21)(22,27)(23,28);(5,8,9,7)(6,11,12,10)(13,16)(14,21)(15,25)(17,26)(18,24)(19,20)(22,28)(23,27);(6,12)(10,11)(13,15,17,18)(14,22,23,19)(16,24,26,25)(20,27,28,21)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284940, #72308085, #72308311, … |
| maximal normal subgroup of | #72284940, #72308085, #72308311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394943);