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 | |
9276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(6,7)(10,11)(16,24)(20,25)(22,28)(26,27);(5,6,9,7)(8,10,12,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(2,4)(5,7,9,6)(8,10,12,11)(14,18)(19,21)(20,25)(26,27);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|b^2=g,c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72249462, #72249486, #72249538, … |
| maximal normal subgroup of | #72249462, #72249486, #72249538, … |
| maximal quotient of | #72326860, #72327774, #72329017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7579974);