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 | |
5860 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)(3,7)(4,8)(5,6)(10,13)(14,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32);(2,6)(3,8)(11,15)(14,16)(18,19)(20,27)(24,26)(28,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(18,19)(23,25)(24,26)(30,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249491, #72249501, #72249524, … |
| maximal normal subgroup of | #72249491, #72249501, #72249524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021531);