C₂×#7685379 #72304272

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7685379

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

88805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234464
maximal normal subgroup of	C₂×#72234464

GAP

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

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

C2×#7685379 #72304272