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 | |
9050 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)(7,8)(9,11)(10,12)(16,24)(19,26)(23,27)(25,28);(1,2)(3,4)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27);(5,7)(6,10)(8,11)(9,12)(13,15,18,20)(14,17,21,22)(16,23,24,27)(19,25,26,28);(13,16)(14,19)(15,23)(17,25)(18,24)(20,27)(21,26)(22,28);(5,8,9,10)(6,7,11,12)(13,14)(15,17)(16,19)(18,21)(20,22)(23,25)(24,26)(27,28);(5,7)(6,8)(9,12)(10,11)(13,17)(14,20)(15,21)(16,25)(18,22)(19,27)(23,26)(24,28)Polycyclic presentation:
2^9|c^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72249462, #72249498, #72249522 |
| maximal normal subgroup of | #72249462, #72249498, #72249522 |
| maximal quotient of | #72326866, #72326889, #72327803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7563772);