C₂×#1418361 #72331131

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#1418361

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

67681 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72252054, C₂×#72261877, C₂×#72261284
maximal normal subgroup of	C₂×#72252054, C₂×#72261877, C₂×#72261284

GAP

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

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

C2×#1418361 #72331131