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 | |
7175 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)(7,9)(8,11)(10,12)(13,14)(15,24)(16,26)(17,27)(18,23)(19,20)(21,25)(22,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,20)(15,25)(18,26)(22,27);(5,6)(7,10)(8,11)(9,12)(13,15,19,25)(14,21,20,24)(16,27,23,22)(17,26,28,18);(5,7)(6,10)(8,12)(9,11)(13,16,19,23)(14,18,20,26)(15,22,25,27)(17,21,28,24);(5,7)(6,9)(8,12)(10,11)(13,17,19,28)(14,22,20,27)(15,18,25,26)(16,21,23,24);(5,6)(7,10)(8,11)(9,12)(13,18)(14,23)(15,17)(16,20)(19,26)(21,22)(24,27)(25,28)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=h,e^2=h,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7874144);