C₂×#3962384 #72325184

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

Aliases

direct product

C₂×#3962384

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

38987 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232089, C₂×#72232165, C₂×#72232119
maximal normal subgroup of	C₂×#72232089, C₂×#72232165, C₂×#72232119

GAP

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

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

C2×#3962384 #72325184