Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,19,21,12)(9,22)(10,14)(11,18)(13,23)(16,29)(17,30,31,36)(20,27,24,28)(25,34,35,33)(26,32);(1,2)(3,4)(5,7,9,18,10,16,23,32)(6,11,13,15,14,26,22,29)(8,17,24,33,21,31,20,34)(12,25,28,30,19,35,27,36);(2,4)(7,16)(8,20)(9,23)(12,27)(13,22)(15,29)(17,33)(19,28)(21,24)(25,36)(30,35)(31,34);(5,8)(6,12)(7,17)(9,24)(10,21)(11,25)(13,28)(14,19)(15,30)(16,31)(18,33)(20,23)(22,27)(26,35)(29,36)(32,34)Polycyclic presentation:
2^9|a^2=f,b^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285452, #72285552, #72285620, … |
| maximal normal subgroup of | #72285452, #72285552, #72285620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363836);