C₂×#647738 #72323912

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

Aliases

direct product

C₂×#647738

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41177 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)(11,12)(13,17)(14,15)(16,18)(19,20,24,25)(21,22,23,26);(8,10)(12,14)(16,17)(20,25)(21,23);(7,8)(9,10)(19,21,24,23)(20,26,25,22);(3,4)(5,6)(11,13)(12,16)(14,17)(15,18)(19,22)(20,21)(23,25)(24,26);(1,2);(3,4)(5,6)(11,14,15,12)(13,17,18,16)(19,23,24,21)(20,22,25,26)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323245, C₂×#72323270, C₂×#72323261
maximal normal subgroup of	C₂×#72323245, C₂×#72323270, C₂×#72323261

GAP

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

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

C2×#647738 #72323912