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 | |
7716 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)(7,12)(8,10)(9,11)(13,14)(15,20)(16,18)(17,25)(19,21)(22,26)(23,27)(24,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,16)(14,20,21,18)(17,24,26,23)(22,27,25,28);(6,11)(10,12)(14,21)(15,16)(17,26)(27,28);(5,6)(7,10)(8,12)(9,11)(13,16,19,15)(14,18,21,20)(17,24,26,23)(22,27,25,28);(5,8,9,7)(6,12,11,10)(13,17)(14,22)(15,23)(16,24)(18,27)(19,26)(20,28)(21,25);(13,18)(14,15)(16,21)(17,27)(19,20)(22,23)(24,25)(26,28)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72323293, C2×#7719290 |
| maximal normal subgroup of | #72323293, C2×#7719290 |
| maximal quotient of | #72323333, #72323343, #72323409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626486);