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 | |
8992 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)(7,11)(10,12)(14,19)(16,25)(17,27)(20,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,20,24,23)(16,21,25,26)(17,22,27,28);(6,9)(10,12)(13,14)(15,23)(16,26)(17,22)(18,19)(20,24)(21,25)(27,28);(13,15)(14,20)(16,17)(18,24)(19,23)(21,22)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,17)(18,25)(19,26)(20,22)(23,28)(24,27);(5,7)(6,10)(8,11)(9,12)(13,17)(14,22)(15,16)(18,27)(19,28)(20,21)(23,26)(24,25)Polycyclic presentation:
2^9|b^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72267573, #72267581, #72267582, … |
| maximal normal subgroup of | #72267573, #72267581, #72267582, … |
| maximal quotient of | #72335350, #72335438, #72335459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615569);