Finite Group Databasebeta

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Cayley diagram

C2×#24148 #6448658

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24148

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2002C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6448661, C2×#6453874, C2×#6453878
maximal normal subgroup ofC2×#6448661, C2×#6453874, C2×#6453878
maximal quotient ofC2×#258135, #72309270

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355854);