Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,19,23,30)(8,25)(9,28)(10,18)(11,16)(12,17)(13,21,32,20)(14,26)(15,27)(22,34,24,35)(29,33,36,31);(5,7)(6,13,17,32)(8,20)(9,29)(10,24)(11,22)(12,23)(14,30,28,19)(15,34,25,35)(16,33,18,31)(21,27)(26,36);(2,4)(5,8,11,9,12,27,10,26)(6,14,18,15,17,28,16,25)(7,20,22,29,23,21,24,36)(13,30,31,34,32,19,33,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249929, #72249959, #72249974, … |
| maximal normal subgroup of | #72249929, #72249959, #72249974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54439);