Finite Group Databasebeta

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Cayley diagram

C2×#29083 #6460595

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27

Aliases

direct productC2×#29083

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2016 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)(13,17,18,15);(4,7)(8,9)(11,13,14,18)(12,15,16,17)(19,20,23,24,22,25,21,26);(12,16)(15,17)(20,24)(21,23)(25,26);(3,5,6,10)(4,8,7,9)(12,16)(15,17)(20,25)(24,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6742086, C2×#6460700, C2×#6699796
maximal normal subgroup ofC2×#6742086, C2×#6460700, C2×#6699796

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367791);