Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196110 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2527 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)(7,12,18,27)(8,19)(9,11)(10,15)(13,20)(14,22)(16,30,32,36)(17,24,33,26)(21,29)(23,28)(25,34,35,31);(1,2)(3,4)(6,11)(9,22)(12,24)(13,19)(14,15)(17,33)(20,23)(25,30)(26,27)(28,29)(31,34)(35,36);(2,4)(5,7)(6,12,15,27)(8,16)(9,17)(10,18)(11,24,14,26)(13,25,28,35)(19,30,29,36)(20,31)(21,32)(22,33)(23,34);(5,8,10,21)(6,13,15,28)(7,16,18,32)(9,23,22,20)(11,19,14,29)(12,25,27,35)(17,34,33,31)(24,30,26,36)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286356, #72286940, #72287104 |
| maximal normal subgroup of | #72286356, #72286940, #72287104 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281311);