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 | |
7028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(11,15)(12,16)(18,23)(19,24);(2,5)(6,8)(9,10,13,14)(11,16,15,12)(17,18,22,23)(19,20,24,21);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24);(2,5)(6,8)(9,10,13,14)(11,16,15,12)(17,20,22,21)(18,24,23,19);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,21,22,20)(18,19,23,24)Polycyclic presentation:
2^9|b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72241084, #72241102, #72241103, … |
| maximal normal subgroup of | #72241084, #72241102, #72241103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538200);