C₂×#686055 #72324459

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#686055

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

51897 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232037, C₂×#72232036
maximal normal subgroup of	C₂×#72232037, C₂×#72232036

GAP

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

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

C2×#686055 #72324459