Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1851 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)(5,6)(7,17)(8,22,25,32)(9,12)(10,15)(11,16)(13,20)(14,26,31,24)(18,33,34,28)(19,29)(21,27)(23,30,35,36);(6,12)(9,10)(13,27)(14,22)(15,16)(17,29)(19,20)(23,35)(24,26)(28,36)(30,33)(31,32);(2,4)(5,7,11,21)(6,13,16,29)(8,23,25,35)(9,20,10,19)(12,27,15,17)(14,30,31,36)(18,26,34,24)(22,28,32,33);(5,8)(6,14)(7,18)(9,26)(10,24)(11,25)(12,22)(13,28)(15,32)(16,31)(17,30)(19,35)(20,23)(21,34)(27,36)(29,33)Polycyclic presentation:
2^9|a^2=ef,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318925, #72318935, #72318947 |
| maximal normal subgroup of | #72318925, #72318935, #72318947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361873);