Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,8,4,6)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(10,14)(11,16)(19,26)(22,25)(27,32)(29,31);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(18,23)(22,25)(24,30)(29,31);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279564, #72280721, #72281380, … |
| maximal normal subgroup of | #72279564, #72280721, #72281380, … |
| maximal quotient of | #72280228, #72303407, #72304083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611305);