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 | |
8866 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)(13,14)(15,19)(16,21)(17,25)(18,20)(22,26)(23,28)(24,27);(5,7)(6,10)(8,11)(9,12)(13,15,18,16)(14,19,20,21)(17,23,26,24)(22,27,25,28);(2,4)(5,8,9,6)(7,11,12,10)(14,20)(15,16)(22,25)(23,24);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,15)(14,21,20,19)(17,24,26,23)(22,28,25,27);(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,28)(25,26);(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,25)(21,28)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,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, #72302931, #72342306, … |
| maximal normal subgroup of | #72302788, #72302931, #72342306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7578474);