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 | |
5422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,8)(6,7)(10,13)(14,16)(17,18)(19,20)(21,29)(22,23)(24,30)(25,27)(26,32)(28,31);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(18,23)(20,27)(24,29)(28,32);(1,2,5,8)(3,4,7,6)(9,11,12,15)(10,14,13,16)(18,23)(19,25)(24,29)(26,31);(1,3)(2,6)(4,8)(5,7)(17,19)(18,20)(21,26)(22,25)(23,27)(24,28)(29,32)(30,31);(1,3)(2,4)(5,7)(6,8)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32);(1,4)(2,7)(3,8)(5,6)(17,21)(18,24)(19,26)(20,28)(22,30)(23,29)(25,31)(27,32)Polycyclic presentation:
2^9|c^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72326379, #72334246 |
| maximal normal subgroup of | #72326379, #72334246 |
| maximal quotient of | #72326438, #72326762, #72327757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944124);