Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417558 |
| 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 | |
2457 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)(11,25)(15,29)(26,36)(31,32);(5,6)(7,15)(8,12)(9,21)(10,13)(11,18)(14,24)(16,32)(17,29)(19,30)(20,27)(22,25)(23,28)(26,34)(31,33)(35,36);(5,7)(6,11)(8,16)(9,22)(10,17)(12,26)(13,25)(14,29)(15,21)(18,24)(19,35)(20,33)(23,34)(27,36)(28,31)(30,32);(2,4)(5,8,10,20)(6,12,13,27)(7,16,17,33)(9,23,24,19)(11,26,25,36)(14,30,21,28)(15,31,29,32)(18,35,22,34)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72312574, #72312585, #72312596 |
| maximal normal subgroup of | #72312574, #72312585, #72312596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324755);