Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161854 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | #35238 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,13,10,14,8,12)(7,15,18,19,16,20,17,11)(21,22)(23,26,29,32)(24,27)(25,28)(30,33,35,36)(31,34);(5,7)(6,11,14,19)(8,17)(9,18)(10,16)(12,20,13,15)(21,23,25,29)(22,26)(24,30,31,35)(27,33)(28,32)(34,36);(1,2,3,4)(7,16)(11,19)(15,20)(17,18)(23,29)(24,31)(26,32)(27,34);(5,8,10,9)(6,12,14,13)(7,17,16,18)(11,20,19,15)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112231);