C₂×#648547 #72341675

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

Aliases

direct product

C₂×#648547

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

56293 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)(5,6)(7,8)(9,10)(11,12,14,13)(15,16)(17,20)(18,19)(21,22);(3,5,6,4)(7,9,10,8)(12,13)(16,19)(17,21);(4,5)(7,9,10,8);(3,5,6,4)(7,9,10,8)(11,13)(12,14)(15,16,18,19)(17,22,21,20);(11,13)(12,14)(15,17,18,21)(16,20,19,22);(3,5)(4,6)(7,9)(8,10)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72268519, C₂×#72268518, C₂×#120×C₂²⋊Q₈
maximal normal subgroup of	C₂×#72268519, C₂×#72268518, C₂×#120×C₂²⋊Q₈

GAP

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

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

C2×#648547 #72341675