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 | |
9015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,22)(17,20)(18,19)(23,25)(24,28)(26,27);(1,2)(3,4)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,27)(26,28);(5,6,8,10)(7,9,12,11)(14,19)(15,22)(20,27)(23,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,27)(26,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,25)(18,21)(19,22)(20,28)(23,27)(24,26);(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)Polycyclic presentation:
2^9|c^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72326381, #72326449, #72326660, … |
| maximal normal subgroup of | #72326381, #72326449, #72326660, … |
| maximal quotient of | #72326394, #72326410, #72326473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7810609);