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 | |
8879 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,18,21,25)(16,19,24,28)(22,26,23,27);(1,2)(3,4)(13,15)(14,18)(16,23)(17,21)(19,27)(20,25)(22,24)(26,28);(5,6,8,10)(7,9,12,11)(15,21)(18,25)(22,23)(26,27);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,23)(17,21)(19,27)(20,25)(22,24)(26,28);(13,16,17,24)(14,19,20,28)(15,22,21,23)(18,26,25,27)Polycyclic presentation:
2^9|a^2=i,c^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72236842, #72238148, #72238189, … |
| maximal normal subgroup of | #72236842, #72238148, #72238189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7890456);