C₂×#498507 #72313920

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#498507

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33605	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72264735, C₂×#72265085, C₂×#72265177
maximal normal subgroup of	C₂×#72264735, C₂×#72265085, C₂×#72265177

GAP

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

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

C2×#498507 #72313920