Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(7,11)(14,17)(15,19)(21,22)(23,27)(24,25)(26,28);(2,4)(5,6,9,8)(7,10,11,12)(13,14,16,17)(15,18,19,20)(22,25)(26,27);(1,2)(3,4)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20);(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^9|b^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72297720, #72297724, #72297772, … |
| maximal normal subgroup of | #72297720, #72297724, #72297772, … |
| maximal quotient of | #72298470, #72329836, #72335945 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 757745);