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 | |
9084 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)(8,10)(9,12)(17,25)(21,26)(22,28)(24,27);(1,2)(3,4)(13,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,26)(27,28);(5,6,8,10)(7,9,12,11)(14,19)(16,20)(21,26)(24,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,26)(25,28);(5,7)(6,9)(8,12)(10,11)(13,16,18,20)(14,15,19,23)(17,24,25,27)(21,22,26,28);(13,17)(14,21)(15,22)(16,24)(18,25)(19,26)(20,27)(23,28)Polycyclic presentation:
2^9|c^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249462, #72249491, #72249543, … |
| maximal normal subgroup of | #72249462, #72249491, #72249543, … |
| maximal quotient of | #72262664, #72262675, #72326873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621719);