C₂×#1260442 #72294752

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#1260442

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

72187 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293914, C₂×#72250996, C₂×#72233704, …
maximal normal subgroup of	C₂×#72293914, C₂×#72250996, C₂×#72233704, …
maximal quotient of	C₂×#72293947, C₂×#72278217, C₂×#72251034

GAP

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

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

C2×#1260442 #72294752