C₂×#1260122 #72332767

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

Aliases

direct product

C₂×#1260122

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41807 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)(19,20)(21,25)(22,24)(23,26);(3,4)(5,6)(7,8)(9,10)(11,12,15,13)(14,16,18,17)(21,26)(23,25);(3,4)(5,6)(19,20,22,24)(21,25,26,23);(1,2);(3,4,5,6)(11,13,15,12)(14,16,18,17)(19,20)(21,25)(22,24)(23,26);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,26)(24,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323394, C₂×#72323407, C₂×#72323388
maximal normal subgroup of	C₂×#72323394, C₂×#72323407, C₂×#72323388

GAP

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

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

C2×#1260122 #72332767