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 | |
7498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,18,20)(15,19,22,17)(16,21,25,27)(23,26,28,24);(1,2,3,4)(13,15)(14,19)(16,23)(17,20)(18,22)(21,26)(24,27)(25,28);(6,10)(7,11)(14,20)(15,22)(21,27)(23,28);(5,7,9,11)(6,8,10,12)(13,16,18,25)(14,21,20,27)(15,23,22,28)(17,24,19,26);(5,8)(6,11)(7,10)(9,12)(13,15)(14,17)(16,23)(18,22)(19,20)(21,24)(25,28)(26,27);(13,17)(14,15)(16,24)(18,19)(20,22)(21,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238230, #72238235, #72238250 |
| maximal normal subgroup of | #72238230, #72238235, #72238250 |
| maximal quotient of | #72263517 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7641582);