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 | |
5178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,8,7)(9,10,13,15)(11,12,16,14)(17,18,23,26)(19,24,20,22)(21,25,30,32)(27,31,28,29);(2,6)(4,7)(9,11,13,16)(10,14,15,12)(19,20)(22,24)(27,28)(29,31);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15)(17,19,23,20)(18,24,26,22)(21,28,30,27)(25,29,32,31);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,20,23,19)(18,22,26,24)(21,27,30,28)(25,31,32,29);(1,3)(2,4)(5,8)(6,7)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(1,4)(2,3)(5,7)(6,8)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gi,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72341056 |
| maximal normal subgroup of | #72341056 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270912);