C₂×#4312903 #72331786

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

Aliases

direct product

C₂×#4312903

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

45745 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)(12,15)(16,18)(19,20,22,24)(21,25,26,23);(4,6)(7,8)(9,10)(13,17)(16,18)(19,20,22,24)(21,23,26,25);(11,12,14,15)(13,18,17,16)(19,20,22,24)(21,25,26,23);(1,2);(3,4)(5,6)(11,12)(13,18)(14,15)(16,17)(19,20)(21,25)(22,24)(23,26);(11,13)(12,16)(14,17)(15,18)(19,21)(20,23)(22,26)(24,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323324, C₂×#72323308, C₂×#72323302
maximal normal subgroup of	C₂×#72323324, C₂×#72323308, C₂×#72323302

GAP

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

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

C2×#4312903 #72331786