Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#577 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,11)(8,12)(9,14)(10,13)(16,33,34,25)(17,31,22,26)(19,36,35,27)(20,29)(21,28)(23,32)(24,30);(5,7)(6,11)(8,19)(9,22)(10,18)(12,27)(13,15)(14,31)(16,20)(17,24)(21,35)(23,34)(25,29)(26,30)(28,36)(32,33);(2,4)(5,8,10,21)(6,12,13,28)(7,16,18,34)(9,23,24,20)(11,25,15,33)(14,32,30,29)(17,35,22,19)(26,36,31,27);(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,15)(14,26)(19,20)(21,34)(22,24)(23,35)(27,29)(28,33)(30,31)(32,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255817, #72262403, #72262421, … |
| maximal normal subgroup of | #72255817, #72262403, #72262421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266807);