Finite Group Databasebeta

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Cayley diagram

C2×#24803 #6443041

order512
nilpotency class3
isoclinismisoclinic class of #24802
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24803

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2748 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);(7,8,12,17,13,10,20,16)(9,18,11,14,19,15,22,21);(3,4)(5,6)(7,9,12,11,13,19,20,22)(8,14,16,18,10,21,17,15);(4,6)(7,10)(8,13)(9,14)(11,18)(12,17)(15,22)(16,20)(19,21);(7,11,20,9,13,22,12,19)(8,15,17,21,10,18,16,14)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72322772, #72322782, #72322798, …
maximal normal subgroup of#72322772, #72322782, #72322798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350237);