C₂×#635986 #72331118

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂×#635986

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

95077 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)(12,14)(15,16,18,20)(17,21,22,19);(8,10)(11,12)(13,14)(15,16,18,20)(17,19,22,21);(8,10);(1,2);(3,4)(5,6)(7,8,9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,22);(15,17)(16,19)(18,22)(20,21)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323281, C₂×#72240166, C₂×#72256575, …
maximal normal subgroup of	C₂×#72323281, C₂×#72240166, C₂×#72256575, …
maximal quotient of	C₂×#72298421, C₂×#72334313, #72334975

GAP

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

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

C2×#635986 #72331118