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Cayley diagram

C2×#10943 #354108

order512
nilpotency class3
isoclinismisoclinic class of #471
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#10943

Subgroups

subgroupG/H
center, socleC24#121
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

3434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#379083, #72215962, #72216082, …
maximal normal subgroup ofC2×#379083, #72215962, #72216082, …
maximal quotient of#72215865, #72215897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261304);