Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,34,19,25,18,24)(15,26,28,21,16,22,29,37)(17,23,35,44,33,41,32,40)(27,39,42,36,30,38,31,43);(5,6)(7,11)(8,10)(9,12)(13,15,19,16)(14,21,25,37)(17,31,33,42)(18,28,20,29)(22,34,26,24)(23,39,41,38)(27,35,30,32)(36,44,43,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,15)(14,22,25,26)(17,30,33,27)(18,28,20,29)(21,34,37,24)(23,38,41,39)(31,32,42,35)(36,44,43,40);(7,12)(9,11)(13,17,20,35,19,33,18,32)(14,23,34,44,25,41,24,40)(15,27,28,42,16,30,29,31)(21,36,22,38,37,43,26,39);(6,10)(9,11)(13,18,19,20)(14,24,25,34)(15,28,16,29)(17,32,33,35)(21,22,37,26)(23,40,41,44)(27,42,30,31)(36,38,43,39)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913970);