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 | |
7038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,27)(24,26)(25,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12);(7,11)(8,12)(13,15,18,16)(14,19,21,22)(17,24,25,23)(20,27,28,26);(6,10)(8,12)(13,16,18,15)(14,19,21,22)(17,24,25,23)(20,26,28,27);(5,8)(6,7)(9,12)(10,11)(13,14)(15,19)(16,22)(17,20)(18,21)(23,26)(24,27)(25,28);(5,6)(7,8)(9,10)(11,12)(13,17,18,25)(14,20,21,28)(15,23,16,24)(19,26,22,27)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72282562, #72312899, #72312916, … |
| maximal normal subgroup of | #72282562, #72312899, #72312916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7971274);