Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421159 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,15,9,14,10,23)(7,12,25,28,17,27,18,34)(8,19,24,31,21,13,22,30)(16,32,35,36,33,26,20,29);(2,4)(5,7,9,17)(6,12,14,27)(8,20)(10,18,11,25)(13,29)(15,28,23,34)(16,24)(19,36)(21,35)(22,33)(26,31)(30,32);(1,2,3,4)(5,8,9,21)(6,13)(7,16,17,33)(10,24,11,22)(12,26)(14,19)(15,31)(18,35,25,20)(23,30)(27,32)(28,36)(29,34);(6,14)(10,11)(12,27)(13,19)(18,25)(20,35)(22,24)(26,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ei,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72287261, #72287280, #72287360, … |
| maximal normal subgroup of | #72287261, #72287280, #72287360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342034);