Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| 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 | |
2448 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)(5,6,10,15)(7,16,17,25)(8,21,24,29)(9,11,18,14)(12,26,28,20)(13,23,30,22)(19,27,32,36)(31,34,35,33);(6,11)(7,17)(9,18)(12,25)(13,21)(14,15)(16,28)(19,32)(22,23)(27,35)(29,30)(31,36);(2,4)(5,7,18,26,10,17,9,20)(6,12,11,25,15,28,14,16)(8,19,22,33,24,32,23,34)(13,27,21,35,30,36,29,31);(5,8)(6,13)(7,19)(9,23)(10,24)(11,21)(12,27)(14,29)(15,30)(16,31)(17,32)(18,22)(20,34)(25,35)(26,33)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281299);