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Cayley diagram

C2×#12674 #493145

order512
nilpotency class4
isoclinismisoclinic class of #509
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#12674

Subgroups

subgroupG/H
center, socle, prime radicalC24C2×D8
commutatorC8C22×C42
Frattini, cosocleC22×C8C24
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72210403, #72210543, C2×C4×#2177, …
maximal normal subgroup of#72210403, #72210543, C2×C4×#2177, …
maximal quotient of#72210369, #72210403, C2×#129099, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400341);