C₂×#4315167 #72332565

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#4315167

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2679	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

31353 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)(11,12,15,14)(13,17,18,16)(20,23)(24,26);(4,6)(7,8)(9,10)(11,13,15,18)(12,16,14,17)(21,25)(24,26);(12,14)(16,17)(19,20,22,23)(21,26,25,24);(1,2);(3,4)(5,6)(11,14,15,12)(13,17,18,16)(19,20)(21,26)(22,23)(24,25);(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323417, C₂×#72323415, C₂×#72323399
maximal normal subgroup of	C₂×#72323417, C₂×#72323415, 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)(11,12,15,14)(13,17,18,16)(20,23)(24,26), (4,6)(7,8)(9,10)(11,13,15,18)(12,16,14,17)(21,25)(24,26), (12,14)(16,17)(19,20,22,23)(21,26,25,24), (1,2), (3,4)(5,6)(11,14,15,12)(13,17,18,16)(19,20)(21,26)(22,23)(24,25), (11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26));

C2×#4315167 #72332565