C₂×#147446 #72309745

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#147446

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#102
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

(12,16,17,22)(14,18)(15,21)(19,26,25,23)(20,24);(4,6)(7,8,9,10)(12,17)(16,22)(19,25)(23,26);(3,4,5,6)(8,10)(11,12,18,26,21,25,14,22,13,17,24,23,15,19,20,16);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256851, C₂×#72256927, C₂×#72257059
maximal normal subgroup of	C₂×#72256851, C₂×#72256927, C₂×#72257059
maximal quotient of	#72323022

GAP

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

gap> G := Group((12,16,17,22)(14,18)(15,21)(19,26,25,23)(20,24), (4,6)(7,8,9,10)(12,17)(16,22)(19,25)(23,26), (3,4,5,6)(8,10)(11,12,18,26,21,25,14,22,13,17,24,23,15,19,20,16), (1,2));

C2×#147446 #72309745