C₂×#553146 #72324427

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#553146

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

74417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232035, C₂×#72219264, C₂×#72232039, …
maximal normal subgroup of	C₂×#72232035, C₂×#72219264, C₂×#72232039, …
maximal quotient of	C₂×#72251006, C₂×#72250280, #72337355

GAP

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

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

C2×#553146 #72324427