Finite Group Databasebeta

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Cayley diagram

C2×#26463 #6444591

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#26463

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2630 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);(4,5)(7,8,10,11)(9,13,14,12)(16,19)(20,22);(3,4,6,5)(8,11)(12,13)(16,20)(18,21)(19,22);(3,4,6,5)(15,16,17,19)(18,22,21,20);(3,5,6,4)(7,9,10,14)(8,12,11,13)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×#478, C2×#6578604, C2×#6444783
maximal normal subgroup ofC2×D4×#478, C2×#6578604, C2×#6444783

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351787);