Finite Group Databasebeta

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Cayley diagram

C2×#8125 #181407

order512
nilpotency class3
isoclinismisoclinic class of #8101
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26

Aliases

direct productC2×#8125

Subgroups

subgroupG/H
center, socleC23C2×#95
commutatorC23C22×C42
Frattini, cosocleC2×C42C24
prime radicalC25C42
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238865, #72238868, #72238870, …
maximal normal subgroup of#72238865, #72238868, #72238870, …
maximal quotient of#72238894, #72309110, #72309153

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88603);