C₂×#642354 #72325023

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#642354

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

58309 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232079, C₂×#72232147, C₂×#72232113
maximal normal subgroup of	C₂×#72232079, C₂×#72232147, C₂×#72232113

GAP

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

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

C2×#642354 #72325023