C₂×#712936 #72324418

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂×#712936

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

54341 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)(8,10)(11,12)(13,16)(14,15)(17,18)(19,20,22,25)(21,24,26,23);(4,6)(8,10)(11,13,15,18)(12,16,14,17);(8,10)(13,18)(16,17)(19,21,22,26)(20,23,25,24);(11,13)(12,17)(14,16)(15,18)(19,21)(20,24)(22,26)(23,25);(1,2);(3,4,5,6)(7,8,9,10)(11,14,15,12)(13,17,18,16)(19,20)(21,24)(22,25)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232034, C₂×#72232038, C₂×#72232058
maximal normal subgroup of	C₂×#72232034, C₂×#72232038, C₂×#72232058

GAP

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

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

C2×#712936 #72324418