C₂×#3860442 #72332789

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

Aliases

direct product

C₂×#3860442

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

29889 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323396, C₂×#72323407, C₂×#72323387
maximal normal subgroup of	C₂×#72323396, C₂×#72323407, C₂×#72323387

GAP

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

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

C2×#3860442 #72332789