C₂×#612463 #72341773

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#612463

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

64573 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)(3,4,6,5)(8,10)(12,14)(15,16)(17,20)(18,21)(19,22);(4,5)(7,8,9,10)(11,12,13,14)(15,17,19,18)(16,20,22,21);(8,10)(11,12,13,14)(15,17,19,18)(16,21,22,20);(7,8,9,10)(11,12,13,14)(16,22)(20,21);(7,8,9,10)(11,12,13,14)(15,18,19,17)(16,21,22,20);(3,5,6,4)(7,8)(9,10)(11,12)(13,14)(15,16,19,22)(17,20,18,21)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72303259, C₂×#72303265, C₂×#72260984
maximal normal subgroup of	C₂×#72303259, C₂×#72303265, C₂×#72260984

GAP

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

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

C2×#612463 #72341773