Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300948 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(11,15)(14,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(1,2)(3,7)(4,6)(5,8)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,14)(13,15);(2,6)(3,8)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(2,6)(5,7)(10,14)(13,16)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)Polycyclic presentation:
2^9|d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72283313, #72283363, #72283396, … |
| maximal normal subgroup of | #72283313, #72283363, #72283396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7214192);