C₂×#1394857 #72325235

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

Aliases

direct product

C₂×#1394857

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

65671 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232172, C₂×#72232113, C₂×#72232134
maximal normal subgroup of	C₂×#72232172, C₂×#72232113, C₂×#72232134

GAP

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

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

C2×#1394857 #72325235