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Cayley diagram

C2×#26916 #6439653

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#26916

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7515180, C2×#6439861, C2×#7523178
maximal normal subgroup ofC2×#7515180, C2×#6439861, C2×#7523178

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346849);