C₂×#2032111 #72324974

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#2032111

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

42189 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)(11,12,15,16)(13,14,17,18)(19,20,22,24)(21,23,26,25);(3,4,5,6)(8,10)(12,16)(13,17);(4,6)(13,17)(14,18)(19,21)(20,23)(22,26)(24,25);(4,6)(7,8)(9,10)(11,13,15,17)(12,14,16,18)(19,21)(20,23)(22,26)(24,25);(1,2);(3,4)(5,6)(11,14)(12,13)(15,18)(16,17)(19,20)(21,25)(22,24)(23,26)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232093, C₂×#72232108, C₂×#72232154
maximal normal subgroup of	C₂×#72232093, C₂×#72232108, C₂×#72232154

GAP

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

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

C2×#2032111 #72324974