Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,19,11)(8,20,22,12)(9,14)(10,13)(16,26,18,24)(17,25)(21,33,34,27)(23,28)(29,32)(30,35)(31,36);(5,7)(6,11)(8,17)(9,16)(10,19)(12,25)(13,15)(14,24)(18,23)(20,29)(21,30)(22,32)(26,28)(27,35)(31,34)(33,36);(1,2,3,4)(7,16)(9,23)(11,24)(14,28)(15,26)(17,30)(18,19)(21,34)(25,35)(27,33)(29,36)(31,32);(5,8)(6,12)(7,17)(9,21)(10,22)(11,25)(13,20)(14,27)(15,29)(16,30)(18,31)(19,32)(23,34)(24,35)(26,36)(28,33)Polycyclic presentation:
2^9|a^2=h,c^2=f,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267752);