Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30513 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,8)(9,11)(10,12)(13,14,18,23)(15,24,26,20)(16,28,30,37)(17,22,19,33)(21,29,38,31)(25,40,41,44)(27,36,32,35)(34,42,39,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,26)(14,20,23,24)(16,25,30,41)(17,32,19,27)(21,34,38,39)(22,36,33,35)(28,44,37,40)(29,43,31,42);(2,4)(13,16,18,30)(14,21,23,38)(15,25,26,41)(17,31,19,29)(20,34,24,39)(22,37,33,28)(27,43,32,42)(35,44,36,40);(6,11)(7,12)(14,22)(15,26)(17,19)(20,35)(21,37)(23,33)(24,36)(25,41)(28,38)(29,31)(34,44)(39,40);(5,8,9,10)(6,7,11,12)(13,17,18,19)(14,22,23,33)(15,27,26,32)(16,29,30,31)(20,35,24,36)(21,28,38,37)(25,42,41,43)(34,40,39,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410015);