Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19)(8,22)(9,25,29,31)(10,17)(11,16)(12,18)(13,21)(14,24)(15,28,32,27)(20,34,36,30)(23,35,33,26);(1,2)(3,4)(5,7)(6,13)(8,16)(9,26)(10,14)(11,21)(12,22)(15,23)(17,19)(18,24)(20,27)(25,30)(28,36)(29,35)(31,34)(32,33);(2,4)(5,8,11,24,12,19,10,13)(6,14,17,22,18,21,16,7)(9,23,27,34,29,33,28,30)(15,26,25,36,32,35,31,20);(5,9)(6,15)(7,20)(8,23)(10,28)(11,27)(12,29)(13,30)(14,26)(16,31)(17,25)(18,32)(19,33)(21,35)(22,36)(24,34)Polycyclic presentation:
2^9|a^2=ef,c^2=f,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,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322857 |
| maximal normal subgroup of | #72322857 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331938);