C₂×#1515121 #72297347

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

Aliases

direct product

C₂×#1515121

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

41487 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232107, C₂×#72232183, C₂×#72232168, …
maximal normal subgroup of	C₂×#72232107, C₂×#72232183, C₂×#72232168, …
maximal quotient of	C₂×#72278034

GAP

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

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

C2×#1515121 #72297347