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 | |
9170 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)(15,21)(16,23)(19,25)(20,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,17)(15,19,21,25)(16,20,23,24)(22,26,27,28);(6,9)(10,12)(13,15)(14,19)(16,22)(17,25)(18,21)(20,26)(23,27)(24,28);(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,17,18,14)(15,19,21,25)(16,24,23,20)(22,26,27,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28)Polycyclic presentation:
2^9|b^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72267574, #72267575, #72267584, … |
| maximal normal subgroup of | #72267574, #72267575, #72267584, … |
| maximal quotient of | #72310491, #72330013, #72330049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549558);