C₂×#8081289 #72332235

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

Aliases

direct product

C₂×#8081289

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 10438 conjugacy classes of subgroups; only the 31829 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323408

GAP

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

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

C2×#8081289 #72332235