Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,26,11,13,10,15)(7,16,20,28,19,31,17,12)(8,21,23,30,25,14,24,32)(18,29,34,36,35,33,22,27);(1,2)(3,4)(5,7)(6,12)(8,22)(9,20)(10,17)(11,19)(13,28)(14,29)(15,31)(16,26)(18,23)(21,33)(24,35)(25,34)(27,30)(32,36);(2,4)(6,13)(7,17)(9,10)(12,16)(14,21)(18,34)(19,20)(22,35)(23,24)(27,33)(28,31)(29,36);(5,8,11,25)(6,14,13,21)(7,18,19,35)(9,24,10,23)(12,27,28,36)(15,32,26,30)(16,33,31,29)(17,34,20,22)Polycyclic presentation:
2^9|a^2=ef,d^2=i,e^2=i,g^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,g^c=gi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285177, #72285492, #72285522, … |
| maximal normal subgroup of | #72285177, #72285492, #72285522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326073);