C₂×#2171295 #72297425

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

Aliases

direct product

C₂×#2171295

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

41955 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)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,23)(24,26);(8,10)(12,14)(13,17)(19,21)(20,24)(22,25)(23,26);(13,17)(16,18)(21,26)(24,25);(4,6)(7,8,9,10)(11,13,15,17)(12,16,14,18)(19,21)(20,25)(22,24)(23,26);(3,4,5,6)(7,8)(9,10)(11,14,15,12)(13,18,17,16)(19,22,23,20)(21,24,26,25);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72277937
maximal normal subgroup of	C₂×#72277937
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((4,6)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,23)(24,26), (8,10)(12,14)(13,17)(19,21)(20,24)(22,25)(23,26), (13,17)(16,18)(21,26)(24,25), (4,6)(7,8,9,10)(11,13,15,17)(12,16,14,18)(19,21)(20,25)(22,24)(23,26), (3,4,5,6)(7,8)(9,10)(11,14,15,12)(13,18,17,16)(19,22,23,20)(21,24,26,25), (1,2));

C2×#2171295 #72297425