C₂×#990368 #72331802

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#990368

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

43419 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323334, C₂×#72323300, C₂×#72323306
maximal normal subgroup of	C₂×#72323334, C₂×#72323300, C₂×#72323306

GAP

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

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

C2×#990368 #72331802