C₂×#782878 #72302578

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

Aliases

direct product

C₂×#782878

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

30145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280033, C₂×#72280075, C₂×#72280331
maximal normal subgroup of	C₂×#72280033, C₂×#72280075, C₂×#72280331

GAP

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

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

C2×#782878 #72302578