C₂×#4379766 #72296338

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#4379766

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

50455 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)(13,14)(15,18)(16,17)(20,23)(21,25);(3,4,5,6)(7,8)(9,10)(19,20,22,23)(21,24,25,26);(11,13)(12,16)(14,18)(15,17)(19,20)(21,24)(22,23)(25,26);(3,4)(5,6)(11,13)(12,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,26);(11,14)(12,17)(13,18)(15,16)(19,21)(20,24)(22,25)(23,26);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72277994, C₂×#72277898
maximal normal subgroup of	C₂×#72277994, C₂×#72277898
maximal quotient of	C₂×#72264985, C₂×#72278150

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

C2×#4379766 #72296338