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 | |
9058 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)(8,10)(9,12)(17,25)(21,28)(23,27)(24,26);(1,2)(3,4)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,27)(26,28);(6,10)(9,11)(13,15)(14,19)(16,22)(17,23)(18,20)(21,26)(24,28)(25,27);(5,6)(7,9)(8,10)(11,12)(13,16,18,19)(14,20,22,15)(17,24,25,26)(21,27,28,23);(5,7)(6,9)(8,12)(10,11)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72267574, #72267582, #72336516 |
| maximal normal subgroup of | #72267574, #72267582, #72336516 |
| maximal quotient of | #72335340, #72335429, #72335450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553129);