C₂×#1393209 #72325558

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1393209

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

45363 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232171, C₂×#72232184, C₂×#72232182
maximal normal subgroup of	C₂×#72232171, C₂×#72232184, C₂×#72232182

GAP

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

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

C2×#1393209 #72325558