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Cayley diagram

C2×#34515 #7626145

order512
nilpotency class2
isoclinismisoclinic class of #3532
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC2×#34515

Subgroups

subgroupG/H
center, socleC24C25
commutatorC22C25×C4
Frattini, cosocleC23C26
prime radicalC22×#562C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

7910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×#3534, C2×#7851531
maximal normal subgroup ofC2×C4×#3534, C2×#7851531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533341);