Finite Group Databasebeta

← Prev (#481594) Next (#481596) →

Cayley diagram

C2×#11774 #481595

order512
nilpotency class4
isoclinismisoclinic class of #11753
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#11774

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

1526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,19,21,26)(14,22,24,17)(15,18)(16,20)(23,25);(4,8)(5,6)(9,10)(11,13)(12,16)(14,23)(15,21)(17,26)(18,25)(19,22)(20,24);(3,5,7,6)(4,9,10,8)(14,24)(16,25)(17,22)(20,23)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#482360, #72240371, #72240612, …
maximal normal subgroup ofC2×#482360, #72240371, #72240612, …
maximal quotient of#72261835, #72293122, #72293124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388791);