C₂×#1515122 #72295017

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

Aliases

direct product

C₂×#1515122

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

44457 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)(11,12,15,13)(14,17,18,16)(19,20,23,24)(21,25,26,22);(7,8,9,10)(11,12,15,13)(14,16,18,17)(20,24)(22,25);(4,6)(12,13)(16,17)(19,21,23,26)(20,25,24,22);(11,13,15,12)(14,16,18,17)(19,22,23,25)(20,21,24,26);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,21)(20,22)(23,26)(24,25);(1,2)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293918, C₂×#72232078, C₂×#72232093, …
maximal normal subgroup of	C₂×#72293918, C₂×#72232078, C₂×#72232093, …
maximal quotient of	C₂×#72293936

GAP

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

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

C2×#1515122 #72295017