C₂×#147466 #72309686

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

Aliases

direct product

C₂×#147466

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
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 ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5157 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249948, C₂×#72256851, C₂×#72257073
maximal normal subgroup of	C₂×#72249948, C₂×#72256851, C₂×#72257073

GAP

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

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

C2×#147466 #72309686