Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1911 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)(7,17)(8,22)(9,13)(10,15)(11,16)(12,18)(14,24)(19,35)(20,29)(21,28)(23,30)(25,31)(26,32)(27,36)(33,34);(1,2,3,4)(5,7,11,21)(6,12,16,29)(8,23,26,34)(9,20,10,18)(13,17,15,28)(14,30,31,33)(19,25,36,24)(22,27,32,35);(6,13)(7,18)(9,10)(14,22)(15,16)(17,28)(19,34)(20,21)(23,36)(24,25)(30,33)(31,32);(2,4)(5,8,11,26)(6,14,16,31)(7,19,21,36)(9,25,10,24)(12,27,29,35)(13,22,15,32)(17,33,28,30)(18,34,20,23)Polycyclic presentation:
2^9|b^2=ef,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,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285161, #72285492, #72285552, … |
| maximal normal subgroup of | #72285161, #72285492, #72285552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328265);