C₂×#7685407 #72304274

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7685407

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

85409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234464
maximal normal subgroup of	C₂×#72234464

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

C2×#7685407 #72304274