Finite Group Databasebeta

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Cayley diagram

C2×#40840 #7638715

order512
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#40840

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

6448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7638736, C2×#7660569, C2×#7660258
maximal normal subgroup ofC2×#7638736, C2×#7660569, C2×#7660258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545911);