Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,20,12)(8,21)(9,13)(10,16)(11,15)(14,24)(18,32,33,35)(19,28,25,26)(22,29)(23,30)(27,34,31,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,28)(14,27)(15,17)(16,26)(21,35)(22,33)(23,36)(24,34)(29,32)(30,31);(2,4)(6,13,15,16)(8,22)(12,26,17,28)(14,29,30,21)(18,33)(23,24)(27,35,31,32)(34,36);(5,8,10,24,11,22,9,23)(6,14,13,21,15,30,16,29)(7,18,19,34,20,33,25,36)(12,27,28,35,17,31,26,32)Polycyclic presentation:
2^9|a^2=fh,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250007, #72250413, #72250427, … |
| maximal normal subgroup of | #72250007, #72250413, #72250427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322308);