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Cayley diagram

C2×#28422 #6463161

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#28422

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(3,4,5,6)(7,8)(9,13)(10,11)(12,14)(15,16,19,23,18,22,28,21)(17,24,27,30,26,20,25,29);(8,11)(12,13)(15,17,19,27,18,26,28,25)(16,20,23,29,22,24,21,30);(4,6)(16,21)(17,25)(19,28)(20,24)(22,23)(26,27);(7,9,10,14)(8,12,11,13)(16,22)(17,26)(21,23)(25,27)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6463173, C2×#6575506, C2×#6576156
maximal normal subgroup ofC2×#6463173, C2×#6575506, C2×#6576156

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370357);