C₂×#704598 #72332683

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

Aliases

direct product

C₂×#704598

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

41563 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)(11,12,15,18)(13,14,17,16)(21,25)(24,26);(4,6)(7,8)(9,10)(11,13,15,17)(12,16,18,14)(20,23)(24,26);(19,20)(21,24)(22,23)(25,26);(1,2);(3,4)(5,6)(11,13)(12,14)(15,17)(16,18)(19,20)(21,26)(22,23)(24,25);(11,14)(12,17)(13,18)(15,16)(19,21)(20,24)(22,25)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323390, C₂×#72323305, C₂×#72323399
maximal normal subgroup of	C₂×#72323390, C₂×#72323305, C₂×#72323399

GAP

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

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

C2×#704598 #72332683