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 | |
8382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,10)(11,12)(14,19)(17,27)(20,23)(25,28);(5,6)(7,11)(8,12)(9,10)(13,14,18,19)(15,17,21,27)(16,23,26,20)(22,28,24,25);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(14,19)(15,21)(20,23)(22,24);(5,7,9,8)(6,11,10,12)(13,15,18,21)(14,17,19,27)(16,24,26,22)(20,28,23,25);(5,8,9,7)(6,12,10,11)(13,16,18,26)(14,20,19,23)(15,22,21,24)(17,25,27,28);(13,17)(14,21)(15,19)(16,25)(18,27)(20,24)(22,23)(26,28)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72313537, #72313631 |
| maximal normal subgroup of | #72313537, #72313631 |
| maximal quotient of | #72342115 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538235);