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 | |
6288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(18,21)(22,24);(1,2)(3,6)(4,5)(7,8)(17,18)(19,22)(20,21)(23,24);(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16);(3,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,20,23)(18,22,21,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|c^2=h,d^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72232449, #72300248, #72300622, … |
| maximal normal subgroup of | #72232449, #72300248, #72300622, … |
| maximal quotient of | #72232550, #72300300, #72300638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7977430);