Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,17,21,19,15,18,22)(16,24,27,20,26,28,25,23);(2,4)(6,10)(8,11)(13,15,19,14)(16,23,26,20)(17,21,18,22)(24,27,28,25);(5,6)(7,8)(9,10)(11,12)(13,16)(14,20)(15,23)(17,25)(18,27)(19,26)(21,24)(22,28);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,21,15,22)(16,25,26,27)(20,24,23,28);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,22,15,21)(16,25,26,27)(20,28,23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72249712, #72251222, #72256940, … |
| maximal normal subgroup of | #72249712, #72251222, #72256940, … |
| maximal quotient of | #72249777, #72264518, #72305503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524651);