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 | |
6230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,16,12)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(3,7)(5,8)(10,14)(12,15)(18,22)(23,26)(24,29)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(19,25)(20,28)(23,26)(24,29);(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,5)(4,6)(7,8)(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=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249510, #72249545, #72284500, … |
| maximal normal subgroup of | #72249510, #72249545, #72284500, … |
| maximal quotient of | #72326603, #72326701, #72326926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834045);