Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22)(23,31)(24,30)(25,29)(26,28)(27,32)(33,36)(34,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23,25,32)(22,27,29,31)(24,34,26,33)(28,35,30,36);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(22,28)(24,26)(27,35)(29,30)(31,36)(33,34);(6,13)(11,15)(12,14)(19,20)(22,29)(23,32)(28,30)(33,34);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,24,25,26)(22,30,29,28)(23,33,32,34)(27,35,31,36)Polycyclic presentation:
2^9|b^2=i,c^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284541, #72284625, #72307879, … |
| maximal normal subgroup of | #72284541, #72284625, #72307879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6581732);