Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161262 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3522 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13,15,20)(8,12)(9,11)(10,14,18,16)(17,19)(21,22)(23,33)(24,29)(25,31)(26,30)(27,32)(28,34)(35,36);(5,7)(6,10)(8,16)(9,15)(11,18)(12,20)(13,19)(14,17)(21,23,27,24)(22,28,32,36)(25,34,26,35)(29,30,33,31);(2,4)(21,24,27,23)(22,29,32,33)(25,34,26,35)(28,30,36,31);(6,11)(10,18)(12,19)(13,20)(21,25,27,26)(22,30,32,31)(23,34,24,35)(28,33,36,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92260);