C₂×#146344 #72309782

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

direct product

C₂×#146344

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₈	C₂×C₄³
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4173 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256851, C₂×#72250185, C₂×#72249925
maximal normal subgroup of	C₂×#72256851, C₂×#72250185, C₂×#72249925

GAP

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

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

C2×#146344 #72309782