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 | |
7837 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)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11);(5,7,9,10)(6,8,11,12)(14,15)(19,21)(20,22)(26,27);(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26);(5,6)(7,8)(9,11)(10,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,8)(6,10)(7,11)(9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)Polycyclic presentation:
2^9|a^2=gh,c^2=i,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 | #72238190, #72238219, #72238245, … |
| maximal normal subgroup of | #72238190, #72238219, #72238245, … |
| maximal quotient of | #72280784, #72280903, #72281677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7768582);