Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3902 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)(6,12)(7,16)(8,21)(9,10)(15,24)(17,25)(18,33)(19,20)(22,23)(26,29)(27,32)(35,36);(2,4)(5,6)(7,17)(8,14)(9,24)(10,12)(11,15)(13,19)(16,28)(18,32)(20,25)(21,29)(22,26)(23,30)(27,34)(31,36)(33,35);(5,7)(6,13)(8,18)(9,16)(10,19)(11,20)(12,25)(14,27)(15,28)(17,24)(21,31)(22,34)(23,33)(26,35)(29,32)(30,36);(5,8,11,23)(6,14,15,30)(7,18,20,33)(9,21,10,22)(12,26,24,29)(13,27,28,36)(16,31,19,34)(17,32,25,35)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286606, #72286995, #72287020, … |
| maximal normal subgroup of | #72286606, #72286995, #72287020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301324);