Finite Group Databasebeta

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Cayley diagram

C2×#29190 #6463635

order512
nilpotency class3
isoclinismisoclinic class of #24378
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#29190

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6597249, C2×#6469272, C2×#6619828
maximal normal subgroup ofC2×#6597249, C2×#6469272, C2×#6619828

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370831);