Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ 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:
(2,4)(5,6)(7,15,17,11)(8,19)(9,22)(10,13)(12,20)(14,23)(16,29,30,33)(18,25,32,28)(21,34)(24,31,27,36)(26,35);(1,2,3,4)(5,7,9,18,10,17,23,32)(6,11,22,28,13,15,14,25)(8,16,20,31,21,30,35,36)(12,24,34,29,26,27,19,33);(11,15)(24,27)(25,28)(29,33);(5,8,9,20,10,21,23,35)(6,12,14,19,13,26,22,34)(7,16,18,31,17,30,32,36)(11,24,25,33,15,27,28,29)Polycyclic presentation:
2^9|a^2=f,b^2=eh,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249846, #72249848, #72249854, … |
| maximal normal subgroup of | #72249846, #72249848, #72249854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323394);