Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
5158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,31)(30,32);(1,2,5,6)(3,7,4,8)(17,19)(18,23)(20,24)(21,27)(22,25)(26,30)(28,31)(29,32);(10,13)(14,15)(18,24)(19,25)(21,29)(28,30);(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,24)(21,28)(22,23)(26,32)(27,31)(29,30);(1,4,5,3)(2,8,6,7)(17,21,22,29)(18,26,24,31)(19,27,25,32)(20,28,23,30)Polycyclic presentation:
2^9|b^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72311295, #72317636, #72317689, … |
| maximal normal subgroup of | #72311295, #72317636, #72317689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7681155);