Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2479 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)(7,15)(8,12)(9,22)(10,14)(11,18)(13,23)(16,31)(17,26)(19,25)(20,35)(21,29)(24,32)(27,33)(28,36)(30,34);(5,7)(6,11)(8,16)(9,17)(10,19)(12,27)(13,25)(14,26)(15,22)(18,23)(20,32)(21,34)(24,29)(28,30)(31,35)(33,36);(5,8,10,21)(6,12,14,29)(7,16,19,34)(9,20,23,36)(11,24,26,27)(13,28,22,35)(15,30,25,31)(17,32,18,33);(2,4)(7,17)(9,23)(11,25)(13,22)(15,26)(16,32)(18,19)(20,36)(24,31)(27,30)(28,35)(33,34)Polycyclic presentation:
2^9|c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239541, #72256290, #72256521, … |
| maximal normal subgroup of | #72239541, #72256290, #72256521, … |
| maximal quotient of | #72239562, C2×#417672, #72316748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324868);