C₂×#2137949 #72325381

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#2137949

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

43033 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)(11,12,14,15)(13,16,17,18)(19,20,23,24)(21,25,26,22);(3,4,6,5)(8,10)(12,15)(16,18);(4,5)(8,10)(13,17)(16,18)(21,26)(22,25);(7,8,9,10)(11,12)(13,18)(14,15)(16,17)(19,21)(20,22)(23,26)(24,25);(1,2);(3,5,6,4)(11,13)(12,16)(14,17)(15,18)(19,22)(20,21)(23,25)(24,26)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232074, C₂×#72232168, C₂×#72232087, …
maximal normal subgroup of	C₂×#72232074, C₂×#72232168, C₂×#72232087, …

GAP

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

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

C2×#2137949 #72325381