C₂×#686503 #72302500

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

Aliases

direct product

C₂×#686503

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

42513 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280103, C₂×#72279963, C₂×#72280331
maximal normal subgroup of	C₂×#72280103, C₂×#72279963, C₂×#72280331

GAP

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

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

C2×#686503 #72302500