Finite Group Databasebeta

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Cayley diagram

C2×#13292 #495393

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

Aliases

direct productC2×#13292

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

1302 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)(13,19,22,26)(14,23,20,17)(15,18)(16,21)(24,25);(4,7)(8,9)(11,13,20,21,15,22,14,24)(12,16,23,26,18,25,17,19);(3,5,6,10)(4,8,7,9)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311757, #72311761, C2×#497484, …
maximal normal subgroup of#72311757, #72311761, C2×#497484, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402589);