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Cayley diagram

C2×#7946 #181121

order512
nilpotency class3
isoclinismisoclinic class of #7943
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#7946

Subgroups

subgroupG/H
center, socleC23C2×#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

3086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#234946, #72263946, #72263952, …
maximal normal subgroup ofC2×#234946, #72263946, #72263952, …
maximal quotient of#72264637, #72309229, C2×#127232

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88317);