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Cayley diagram

C2×#17672 #6343794

order512
nilpotency class3
isoclinismisoclinic class of #17577
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#17672

Subgroups

subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

3128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,6,10,7,9,15,17)(5,11,12,18,13,8,14,16)(19,20)(21,28,30,33)(22,31)(23,26)(24,27)(25,29,34,32);(3,5,7,13)(4,8,9,11)(6,14,15,12)(10,18,17,16)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28);(4,9)(8,11)(10,17)(16,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#186252, #72238868, #72238870
maximal normal subgroup ofC2×#186252, #72238868, #72238870
maximal quotient of#72238889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250990);