C₂×#153931 #72290843

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#153931

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

163909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72287884, C₂×#72287883, C₂×#72287889, …
maximal normal subgroup of	C₂×#72287884, C₂×#72287883, C₂×#72287889, …
maximal quotient of	#72340111

GAP

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

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

C2×#153931 #72290843