C₂×#1006045 #72299362

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

Aliases

direct product

C₂×#1006045

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

43819 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)(7,8,9,10)(12,14)(15,16)(17,21)(18,19)(20,22);(7,8,9,10)(11,12)(13,14)(15,16,19,18)(17,21,22,20);(4,5)(7,8,9,10)(16,18)(20,21);(7,8)(9,10)(15,17)(16,20)(18,21)(19,22);(3,5,6,4)(15,18,19,16)(17,21,22,20);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297586, C₂×#72297593
maximal normal subgroup of	C₂×#72297586, C₂×#72297593

GAP

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

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

C2×#1006045 #72299362