C₂×#517777 #72332909

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#517777

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41173 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323393, C₂×#72323394, C₂×#72323421
maximal normal subgroup of	C₂×#72323393, C₂×#72323394, C₂×#72323421

GAP

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

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

C2×#517777 #72332909