Finite Group Databasebeta

← Prev (#6443302) Next (#6443304) →

Cayley diagram

C2×#25746 #6443303

order512
nilpotency class3
isoclinismisoclinic class of #24464
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#25746

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3499C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72312395, C2×#6443305
maximal normal subgroup of#72312395, C2×#6443305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350499);