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 | |
8341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6,9,10)(7,8,11,12)(13,14)(15,18)(16,21)(17,20)(19,24)(22,26)(23,28)(25,27);(1,2)(3,4)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28);(6,10)(7,11)(14,15)(19,21)(20,22)(27,28);(5,7)(6,8)(9,11)(10,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27);(5,8)(6,7)(9,12)(10,11)(13,16)(14,19)(15,21)(17,25)(18,24)(20,28)(22,27)(23,26);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=i,f^2=i,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 | #72236840, #72238202, #72238234, … |
| maximal normal subgroup of | #72236840, #72238202, #72238234, … |
| maximal quotient of | C2×#7941013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7848209);