C₂×#7631902 #72298751

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7631902

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

119837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297215
maximal normal subgroup of	C₂×#72297215
maximal quotient of	C₂×#72298086

GAP

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

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

C2×#7631902 #72298751