Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,11)(9,12)(13,23)(14,24)(15,25)(16,30)(17,26)(19,29)(20,22)(27,31)(28,35)(34,36);(1,2)(3,4)(5,6,12,25,10,15,9,11)(7,13,26,35,18,28,17,23)(8,19,22,14,21,24,20,29)(16,31,33,27,30,34,32,36);(2,4)(5,7)(6,13)(8,16)(9,17)(10,18)(11,23)(12,26)(14,27)(15,28)(19,31)(20,32)(21,30)(22,33)(24,34)(25,35)(29,36);(5,8)(6,14)(7,16)(9,22)(10,21)(11,24)(12,20)(13,27)(15,29)(17,33)(18,30)(19,25)(23,34)(26,32)(28,36)(31,35)Polycyclic presentation:
2^9|b^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72251179, #72262411, #72262421, … |
| maximal normal subgroup of | #72251179, #72262411, #72262421, … |
| maximal quotient of | #72258178, #72258197, #72285403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325056);