Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,21)(15,23,24,38)(16,20,28,37)(17,30,32,22)(19,31,33,26)(25,39,27,41)(29,44,43,36)(34,42,35,40);(7,8)(10,11)(13,15)(14,19)(16,27)(17,31)(18,24)(20,35)(21,33)(22,38)(23,30)(25,28)(26,32)(29,40)(34,37)(36,39)(41,44)(42,43);(2,4)(5,7,9,8)(6,10,12,11)(15,24)(19,33)(23,38)(25,27)(26,31)(34,35)(39,41)(40,42);(5,8,9,7)(6,11,12,10)(13,16)(14,20)(15,25)(17,29)(18,28)(19,34)(21,37)(22,36)(23,39)(24,27)(26,40)(30,44)(31,42)(32,43)(33,35)(38,41);(6,12)(10,11)(14,21)(16,28)(19,33)(22,30)(23,38)(25,27)(29,43)(40,42)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394671);