Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1918 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)(6,12)(7,16)(8,21)(9,10)(13,17)(15,25)(18,33)(19,20)(22,27)(23,24)(28,36)(32,34);(1,2,3,4)(5,6,9,12,11,15,10,25)(7,17,16,26,20,13,19,29)(8,22,21,30,23,27,24,14)(18,34,35,28,33,36,31,32);(5,7,10,16,11,20,9,19)(6,13,12,26,15,17,25,29)(8,18,21,31,23,33,24,35)(14,28,27,36,30,32,22,34);(2,4)(5,8,11,23)(6,14,15,30)(7,18,20,33)(9,24,10,21)(12,27,25,22)(13,28,17,32)(16,31,19,35)(26,36,29,34)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318385, #72318867, #72318947 |
| maximal normal subgroup of | #72318385, #72318867, #72318947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330140);