Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,12,15,23)(7,16,18,24)(14,19,30,26)(17,32,34,31);(1,2)(3,4)(5,6)(7,13)(8,19)(9,23)(10,12)(11,15)(14,20)(16,29)(17,33)(18,28)(21,30)(22,26)(24,25)(27,32)(31,36)(34,35);(2,4)(5,7)(6,13)(8,17)(9,16)(10,24)(11,18)(12,25)(14,27)(15,28)(19,33)(20,32)(21,31)(22,34)(23,29)(26,35)(30,36);(5,8,11,22)(6,14,15,30)(7,17,18,34)(9,21,10,20)(12,26,23,19)(13,27,28,36)(16,31,24,32)(25,35,29,33);(6,15)(12,23)(13,28)(14,30)(19,26)(25,29)(27,36)(33,35)Polycyclic presentation:
2^9|a^2=fg,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^c=gi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410569);