C₂×#1605671 #72338048

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#1605671

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

44467 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232305, C₂×#72232308, C₂×#72232365
maximal normal subgroup of	C₂×#72232305, C₂×#72232308, C₂×#72232365

GAP

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

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

C2×#1605671 #72338048