Finite Group Databasebeta

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Cayley diagram

C2×#7181 #126791

order512
nilpotency class4
isoclinismisoclinic class of #360
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24

Aliases

direct productC2×#7181

Subgroups

subgroupG/H
centerC2×C8#102
commutatorC8C2×C4×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1279
prime radicalC24#104
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226839, #72226843, #72226846, …
maximal normal subgroup of#72226839, #72226843, #72226846, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33987);