C₂×#4306360 #72325593

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

Aliases

direct product

C₂×#4306360

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

62843 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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232148, C₂×#72232161, C₂×#72232170
maximal normal subgroup of	C₂×#72232148, C₂×#72232161, C₂×#72232170

GAP

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

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

C2×#4306360 #72325593