Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1422 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)(7,12,19,27)(8,21)(9,16)(10,15)(11,14)(13,23)(17,28,20,25)(18,32,34,36)(22,29)(24,30)(26,33,35,31);(1,2)(3,4)(5,7)(6,12,14,27)(8,18)(9,17)(10,20)(11,19)(13,26,29,35)(15,25,16,28)(21,32,30,36)(22,33)(23,31)(24,34);(2,4)(7,17)(9,10)(12,25)(13,29)(15,16)(18,33)(19,20)(22,23)(26,32)(27,28)(31,34)(35,36);(5,8,11,24)(6,13,14,29)(7,18,19,34)(9,23,10,22)(12,26,27,35)(15,21,16,30)(17,31,20,33)(25,32,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286641, #72286940, #72287020 |
| maximal normal subgroup of | #72286641, #72286940, #72287020 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363855);