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 | |
7456 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)(13,14,19,18)(15,22,17,20)(16,21,26,25)(23,28,24,27);(1,2)(3,4)(5,7,11,9)(6,8,12,10);(5,8,11,10)(6,9,12,7)(13,15,19,17)(14,20,18,22)(16,24,26,23)(21,28,25,27);(5,7,11,9)(6,8,12,10)(13,16,19,26)(14,21,18,25)(15,23,17,24)(20,27,22,28);(5,9,11,7)(6,8,12,10)(13,17,19,15)(14,22,18,20)(16,24,26,23)(21,28,25,27);(5,10,11,8)(6,9,12,7)(13,18,19,14)(15,22,17,20)(16,25,26,21)(23,28,24,27)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72238247, #72238248, #72238258, … |
| maximal normal subgroup of | #72238247, #72238248, #72238258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7766497);