Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12);(5,7,9,8)(6,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(1,2,3,4)(5,7,9,8)(6,11,12,10)(14,18)(19,24)(20,25)(26,28);(5,8,9,7)(6,10,12,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(15,21)(19,24)(22,27)(26,28);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=g,f^2=i,b^a=bg,c^a=ch,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72279663, #72280047, #72303252, … |
| maximal normal subgroup of | #72279663, #72280047, #72303252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536898);