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Cayley diagram

C2×#33373 #7622481

order512
nilpotency class4
isoclinismisoclinic class of #33359
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#33373

Subgroups

subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C24×C4
Frattini, cosocleC2×Q8C25
prime radicalC2×#3498C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1

4572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4,8,14,9,15,5,11)(6,12,17,10,7,13,16,18);(3,5,9,8)(4,10,15,18)(6,16,7,17)(11,13,14,12);(3,5,9,8)(4,11,15,14)(6,17,7,16)(10,13,18,12);(3,6,9,7)(4,12,15,13)(5,16,8,17)(10,11,18,14)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#152745, #72340642, #72340647
maximal normal subgroup ofC2×#152745, #72340642, #72340647
maximal quotient ofC2×#496222

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529677);