Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1879 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)(7,19)(8,25)(9,29)(10,17)(11,16)(12,18)(13,22)(14,28)(15,27)(20,36)(21,31)(23,33)(24,32)(26,35)(30,34);(1,2,3,4)(5,7)(6,13)(8,26)(9,30)(10,23)(11,22)(12,24)(14,34)(15,35)(16,32)(17,19)(18,33)(20,27)(21,28)(25,31)(29,36);(5,8,12,9)(6,14,18,15)(7,20,24,21)(10,28,11,27)(13,31,33,36)(16,25,17,29)(19,34,32,35)(22,30,23,26);(2,4)(5,9,12,8)(6,15,18,14)(7,21,24,20)(10,27,11,28)(13,31,33,36)(16,29,17,25)(19,34,32,35)(22,26,23,30)Polycyclic presentation:
2^9|b^2=ef,c^2=i,d^2=i,e^2=i,f^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286567, #72287028, #72287089 |
| maximal normal subgroup of | #72286567, #72287028, #72287089 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328267);