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Cayley diagram

C2×#7178 #126784

order512
nilpotency class4
isoclinismisoclinic class of #360
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C25

Aliases

direct productC2×#7178

Subgroups

subgroupG/H
center, socle, prime radicalC24#102
commutatorC8C2×C4×C8
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×#1333, C2×#128688, C2×#129165, …
maximal normal subgroup ofC2×C4×#1333, C2×#128688, C2×#129165, …
maximal quotient ofC2×#93270, C2×#94059, C2×#94057, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33980);