C₂×#2078559 #72337974

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

Aliases

direct product

C₂×#2078559

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

47059 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)(7,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,23)(24,26);(12,15)(13,17)(19,20,22,23)(21,25,26,24);(3,4)(5,6)(7,8)(9,10)(12,15)(13,17)(20,23)(21,26);(4,6)(8,9)(11,12)(13,18)(14,15)(16,17)(19,20)(21,24)(22,23)(25,26);(1,2);(3,4)(5,6)(7,9,10,8)(11,13,14,17)(12,16,15,18)(19,21,22,26)(20,24,23,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232354, C₂×#72232311, C₂×#72232298
maximal normal subgroup of	C₂×#72232354, C₂×#72232311, C₂×#72232298

GAP

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

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

C2×#2078559 #72337974