C₂×#3332799 #72337894

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#3332799

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

49759 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:

(7,8,10,9)(11,12)(13,15)(14,17)(16,18)(19,20,23,21)(22,25,26,24);(11,12,15,13)(14,16,18,17)(19,20)(21,23)(22,25)(24,26);(4,6)(8,9)(12,13)(14,18)(20,21)(22,26);(4,6)(7,8,10,9)(11,13,15,12)(14,16,18,17)(19,21,23,20)(22,24,26,25);(1,2);(3,4,5,6)(7,9,10,8)(11,14,15,18)(12,16,13,17)(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×#72232318, C₂×#72232305, C₂×#72232314
maximal normal subgroup of	C₂×#72232318, C₂×#72232305, C₂×#72232314

GAP

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

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

C2×#3332799 #72337894