C₂×#2127723 #72331787

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

Aliases

direct product

C₂×#2127723

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

40327 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72317974, C₂×#72297936, C₂×#72323295
maximal normal subgroup of	C₂×#72317974, C₂×#72297936, C₂×#72323295

GAP

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

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

C2×#2127723 #72331787