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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(14,19)(15,22)(16,24)(18,26)(20,25)(23,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,23)(17,19)(18,21)(22,24)(25,27)(26,28);(2,4)(13,15)(14,20)(16,19)(17,23)(18,27)(21,25)(22,26)(24,28);(6,10)(7,12)(13,16,17,20)(14,15,19,23)(18,24,26,25)(21,27,28,22);(14,19)(15,23)(21,28)(22,27)Polycyclic presentation:
2^9|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,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249595, #72249615, #72249660 |
| maximal normal subgroup of | #72249595, #72249615, #72249660 |
| maximal quotient of | #72264490, #72307868, #72308878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407907);