Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1277 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)(5,6,9,13,10,14,22,29)(7,15,18,25,19,24,17,11)(8,12,21,28,20,27,35,36)(16,31,32,23,33,30,34,26);(5,7)(6,11)(8,16)(9,17)(10,19)(12,26)(13,24)(14,25)(15,29)(18,22)(20,33)(21,34)(23,27)(28,30)(31,36)(32,35);(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,28)(14,27)(15,30)(17,34)(18,32)(19,33)(22,35)(24,31)(25,26)(29,36);(2,4)(7,17,19,18)(8,20)(11,24,25,15)(12,27)(16,32,33,34)(21,35)(23,30,26,31)(28,36)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285522, #72285534, #72285552, … |
| maximal normal subgroup of | #72285522, #72285534, #72285552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334590);