Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| 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 | #2535 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1325 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)(7,16,18,30)(8,13)(9,14)(10,23)(11,15)(12,19,26,20)(17,31,33,32)(21,27)(22,29)(24,28)(25,34,36,35);(5,6)(7,16)(8,13)(9,14)(10,23)(11,15)(12,20)(17,32)(18,30)(19,26)(21,27)(22,29)(24,28)(25,34)(31,33)(35,36);(5,7)(6,12)(8,17)(9,20)(10,19)(11,18)(13,25)(14,30)(15,26)(16,23)(21,35)(22,33)(24,34)(27,32)(28,31)(29,36);(2,4)(5,8,9,21,11,22,10,24)(6,13,14,27,15,29,23,28)(7,17,20,35,18,33,19,34)(12,25,30,32,26,36,16,31)Polycyclic presentation:
2^9|a^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 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, 335139);