Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394098 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
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 | |
3919 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)(7,13)(8,21)(9,17)(10,15)(11,16)(12,19)(14,22)(18,26)(20,29)(23,31)(24,27)(25,30)(28,33)(32,35)(34,36);(5,7)(6,12)(8,18)(9,24)(10,19)(11,20)(13,17)(14,26)(15,29)(16,27)(21,35)(22,33)(23,34)(25,32)(28,31)(30,36);(2,4)(5,7)(6,13)(8,18)(9,24)(10,19)(11,20)(12,15)(14,28)(16,29)(17,27)(21,26)(22,33)(23,34)(25,32)(30,35)(31,36);(5,8,11,23)(6,14,16,30)(7,18,20,34)(9,22,10,25)(12,26,27,36)(13,28,29,35)(15,21,17,31)(19,32,24,33)Polycyclic presentation:
2^9|d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261238, #72261630, #72303161, … |
| maximal normal subgroup of | #72261238, #72261630, #72303161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301295);