C₂×#4311739 #72325035

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

Aliases

direct product

C₂×#4311739

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

56347 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232074, C₂×#72232113, C₂×#72232146
maximal normal subgroup of	C₂×#72232074, C₂×#72232113, C₂×#72232146

GAP

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

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

C2×#4311739 #72325035