C₂×#1418485 #72297014

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#1418485

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

43069 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)(20,23)(24,26);(8,10)(11,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,23)(25,26);(4,6)(7,8)(9,10)(11,12,14,16)(13,17,18,15)(19,21,22,25)(20,24,23,26);(4,6)(8,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,24)(22,25)(23,26);(3,4)(5,6)(7,8)(9,10)(11,13)(12,15)(14,18)(16,17)(19,20)(21,24)(22,23)(25,26);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232122, C₂×#72323251, C₂×#72323260, …
maximal normal subgroup of	C₂×#72232122, C₂×#72323251, C₂×#72323260, …
maximal quotient of	C₂×#72278787, C₂×#72278154

GAP

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

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

C2×#1418485 #72297014