C₂×#686390 #72302682

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#686390

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279977, C₂×#72280103, C₂×#72280257, …
maximal normal subgroup of	C₂×#72279977, C₂×#72280103, C₂×#72280257, …
maximal quotient of	C₂×#72278357, C₂²×#686390

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×#686390 #72302682