Finite Group Databasebeta

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Cayley diagram

C2×#13295 #495403

order512
nilpotency class4
isoclinismisoclinic class of #11541
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#13295

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311773, #72311778, C2×#497484, …
maximal normal subgroup of#72311773, #72311778, C2×#497484, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402599);