Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3742 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)(7,16,19,11)(8,20,22,13)(9,15)(10,14)(12,18,28,23)(17,25)(21,34,36,29)(24,26)(27,31)(30,32)(33,35);(2,4)(5,7)(6,11)(8,17)(9,23)(10,19)(12,26)(13,25)(14,16)(15,28)(18,24)(20,30)(21,35)(22,32)(27,34)(29,33)(31,36);(5,7)(6,12)(8,17)(9,23)(10,19)(11,15)(13,27)(14,28)(16,26)(18,24)(20,33)(21,35)(22,32)(25,29)(30,34)(31,36);(2,4)(5,8,10,22)(6,13,14,20)(7,17,19,32)(9,21,24,36)(11,25,16,30)(12,27,28,33)(15,29,26,34)(18,31,23,35)Polycyclic presentation:
2^9|a^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72303070, #72303077, #72303281, … |
| maximal normal subgroup of | #72303070, #72303077, #72303281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266798);