Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(17,25)(20,24)(22,28)(26,27);(1,2)(3,4)(13,14)(15,19)(16,18)(17,20)(21,23)(22,26)(24,25)(27,28);(6,9)(10,11)(14,16)(19,21)(20,24)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27);(5,7)(6,10)(8,12)(9,11)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72249462, #72249498, #72249522, … |
| maximal normal subgroup of | #72249462, #72249498, #72249522, … |
| maximal quotient of | #72263200, #72326842, #72327678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626303);