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 | |
5437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,8,7)(9,10,12,13)(11,15,16,14)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25);(2,6)(4,7)(10,13)(14,15)(19,28)(21,29)(24,32)(25,30);(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32);(1,4)(2,3)(5,7)(6,8)(9,11,12,16)(10,14,13,15)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32);(1,3)(2,4)(5,8)(6,7)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=ghi,c^2=hi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72317943, #72317954, #72318056, … |
| maximal normal subgroup of | #72317943, #72317954, #72318056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400149);