Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,13,11,16,9,15)(7,17,19,33,20,34,24,32)(8,14,22,30,23,31,21,29)(12,25,27,18,28,35,26,36);(1,2,3,4)(5,7)(6,12,16,28)(8,18)(9,24)(10,19)(11,20)(13,26,15,27)(14,17,31,34)(21,25)(22,35)(23,36)(29,33,30,32);(6,13)(9,10)(12,26)(14,30)(15,16)(17,32)(19,24)(21,22)(25,35)(27,28)(29,31)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=eh,f^b=fhi,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286210, #72286216, #72286236, … |
| maximal normal subgroup of | #72286210, #72286216, #72286236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53844);