Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,27,37)(16,29,17,31)(18,33,34,23)(21,32,22,30)(25,39,26,41)(28,44,43,38)(35,42,36,40);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,35)(22,36)(23,38)(24,37)(29,39)(30,40)(31,41)(32,42)(33,44)(34,43);(2,4)(13,16,19,17)(14,21,24,22)(15,25,27,26)(18,32,34,30)(20,35,37,36)(23,31,33,29)(28,42,43,40)(38,41,44,39);(6,10)(7,12)(13,17,19,16)(14,22,24,21)(15,26,27,25)(18,30,34,32)(20,36,37,35)(23,29,33,31)(28,40,43,42)(38,39,44,41);(15,27)(20,37)(25,26)(28,43)(35,36)(38,44)(39,41)(40,42)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407872);