Finite Group Databasebeta

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Cayley diagram

C2×#8076 #181681

order512
nilpotency class3
isoclinismisoclinic class of #327
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#8076

Subgroups

subgroupG/H
centerC2×C42C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1600 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)(5,13)(6,10)(7,11)(8,12)(9,14)(15,18)(16,17)(19,20,22,25,23,26,30,33)(21,27,29,34,28,24,31,32);(3,5,7,15,8,16,6,14)(4,9,10,17,12,18,11,13)(19,21)(20,24,26,27)(22,29)(23,28)(25,32,33,34)(30,31);(3,6,8,7)(4,10,12,11)(5,14,16,15)(9,17,18,13)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240032, #72240039, C2×#109996
maximal normal subgroup of#72240032, #72240039, C2×#109996
maximal quotient of#72240035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88877);