C₂×#7697591 #72304571

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

Aliases

direct product

C₂×#7697591

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

At least 9756 conjugacy classes of subgroups; only the 32109 normal subgroups are shown.

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

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237366
maximal normal subgroup of	C₂×#72237366

GAP

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

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

C2×#7697591 #72304571