C₂×#141800 #72258676

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

direct product

C₂×#141800

Subgroups

	subgroup	G/H
center, socle	C₂³	#1972
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#33748	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1791 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)(5,7)(6,8)(9,10)(12,16)(13,18)(14,22)(17,23)(19,24)(20,25)(21,26);(4,7,8,9)(12,17,19,20)(13,21)(16,23,24,25)(18,26);(1,2);(11,12,18,23,14,20,21,24,15,19,26,25,22,17,13,16)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213975
maximal normal subgroup of	C₂×#72213975

GAP

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

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

C2×#141800 #72258676