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 | |
5826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,8)(5,7)(9,10,13,14)(11,15,16,12)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32);(1,2)(3,7)(4,6)(5,8)(9,10)(11,12)(13,14)(15,16)(18,22)(19,25)(24,29)(27,31);(1,3,4,8)(2,5,6,7)(9,11,13,16)(10,12,14,15)(19,25)(20,28)(23,26)(24,29);(2,6)(5,7)(10,14)(12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(1,3)(2,7)(4,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249374, #72249382, #72249410, … |
| maximal normal subgroup of | #72249374, #72249382, #72249410, … |
| maximal quotient of | #72263132, #72263324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947432);