C₂×#1572498 #72296192

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

Aliases

direct product

C₂×#1572498

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

51067 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)(5,6)(7,8)(9,10)(20,23)(24,26);(4,5)(7,8,10,9)(13,17)(16,18)(19,20,22,23)(21,24,25,26);(3,4,6,5)(7,8,10,9)(12,15)(13,17)(20,23)(21,25);(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,26)(22,23)(24,25);(3,5,6,4)(7,9,10,8)(11,13,14,17)(12,16,15,18)(19,21,22,25)(20,24,23,26);(1,2)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231349, C₂²×#1572498, C₂×#72277898
maximal normal subgroup of	C₂×#72231349, C₂²×#1572498, C₂×#72277898
maximal quotient of	C₂²×#1572498, C₂×#72278121

GAP

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

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

C2×#1572498 #72296192