C₂×#612464 #72341766

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

Aliases

direct product

C₂×#612464

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#525	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

59457 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)(17,22)(18,21)(19,20);(4,5)(16,20)(21,22);(8,10)(11,12,13,14)(15,17,19,18)(16,21,20,22);(7,8,9,10)(11,12,13,14)(16,20)(21,22);(7,8,9,10)(11,12,13,14)(15,18,19,17)(16,21,20,22);(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,22)(18,21)(19,20)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72268526, C₂×#72303259, C₂×#72268529
maximal normal subgroup of	C₂×#72268526, C₂×#72303259, C₂×#72268529

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

C2×#612464 #72341766