C₂×#636416 #72332669

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

Aliases

direct product

C₂×#636416

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

41691 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)(11,12,15,13)(14,16,18,17)(19,20,22,24)(21,23,25,26);(3,4)(5,6)(7,8)(9,10)(11,12,15,13)(14,17,18,16)(19,21,22,25)(20,23,24,26);(19,21)(20,23)(22,25)(24,26);(1,2);(3,4,5,6)(11,13,15,12)(14,17,18,16)(19,21)(20,23)(22,25)(24,26);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,20)(21,23)(22,24)(25,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323396, C₂×#72323305, C₂×#72323406
maximal normal subgroup of	C₂×#72323396, C₂×#72323305, C₂×#72323406

GAP

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

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

C2×#636416 #72332669