Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1376 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)(5,6,11,16,12,18,10,17)(7,19,21,13,22,32,24,33)(8,23,20,14,26,34,25,35)(9,27,28,15,29,31,30,36);(2,4)(5,7,11,21,12,22,10,24)(6,13,16,32,18,33,17,19)(8,9,20,28,26,29,25,30)(14,31,34,36,35,27,23,15);(5,8,11,20,12,26,10,25)(6,14,17,23,18,35,16,34)(7,9,21,28,22,29,24,30)(13,31,19,15,33,27,32,36);(5,9)(6,15)(7,20)(8,24)(10,30)(11,28)(12,29)(13,14)(16,31)(17,27)(18,36)(19,23)(21,26)(22,25)(32,34)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,e^b=ei,g^b=gi,d^c=di,e^d=ei,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307190, #72307196, #72307204, … |
| maximal normal subgroup of | #72307190, #72307196, #72307204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300148);