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 |
| prime radical | #34980 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9086 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)(5,6)(7,11)(8,9)(10,12)(16,24)(19,26)(22,28)(25,27);(5,7)(6,10)(8,11)(9,12)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(2,4)(5,8,9,6)(7,11,12,10)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12);(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)Polycyclic presentation:
2^9|b^2=i,c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72302788, #72302952, #72305286, … |
| maximal normal subgroup of | #72302788, #72302952, #72305286, … |
| maximal quotient of | #72306250 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577862);