C₂×#1258857 #72325585

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1258857

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

42779 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)(11,12)(13,17)(14,15)(16,18)(19,20)(21,24)(22,25)(23,26);(4,6)(7,8)(9,10)(13,18)(16,17)(19,21)(20,22)(23,25)(24,26);(8,10)(12,15)(13,18)(21,25)(22,24);(1,2);(7,8)(9,10)(11,12)(13,17)(14,15)(16,18)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,13,14,18)(12,16,15,17)(19,22,23,24)(20,25,26,21)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232137, C₂×#72232157, C₂×#72232170
maximal normal subgroup of	C₂×#72232137, C₂×#72232157, C₂×#72232170

GAP

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

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

C2×#1258857 #72325585