C₂×#7631291 #72304235

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

Aliases

direct product

C₂×#7631291

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#3496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

92245 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)(9,13)(12,14)(15,16)(17,19)(18,20)(21,22);(7,8)(9,12)(10,11)(13,14)(15,17)(16,19)(18,21)(20,22);(8,11)(12,14)(15,17)(16,19)(18,21)(20,22);(7,9,10,13)(8,12,11,14);(4,6)(9,13)(12,14)(15,18,19,22)(16,20,17,21);(7,9)(8,12)(10,13)(11,14)(15,17)(16,19)(18,21)(20,22);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18)(16,21)(17,20)(19,22)

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234467
maximal normal subgroup of	C₂×#72234467

GAP

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

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

C2×#7631291 #72304235