Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1999 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,20,18)(8,14)(9,25)(10,27)(11,16)(12,19,29,35)(13,23)(15,26)(21,34,36,31)(22,32,28,30)(24,33);(5,7)(6,12)(8,21)(9,19)(10,28)(11,20)(13,32)(14,18)(15,34)(16,29)(17,33)(22,26)(23,35)(24,36)(25,30)(27,31);(2,4)(5,6)(7,18)(8,14)(9,25)(10,27)(11,16)(12,19)(13,23)(15,26)(17,20)(21,31)(22,30)(24,33)(28,32)(29,35)(34,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=dh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237254, #72237285, #72238781, … |
| maximal normal subgroup of | #72237254, #72237285, #72238781, … |
| maximal quotient of | #72237267, #72308760 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55100);