C₂×#856585 #72325155

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#856585

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

49545 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)(13,17)(14,18)(15,16)(19,20)(21,22)(23,25)(24,26);(3,4)(5,6)(7,8)(9,10)(11,13,16,14)(12,17,15,18)(19,21,23,26)(20,24,25,22);(4,5)(7,8)(9,10)(12,15)(17,18)(20,25)(22,24);(4,5)(8,10)(11,14,16,13)(12,17,15,18)(19,21)(20,24)(22,25)(23,26);(1,2);(3,5,6,4)(7,8)(9,10)(11,15,16,12)(13,18,14,17)(19,22,23,24)(20,26,25,21)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232090, C₂×#72232175, C₂×#72232123, …
maximal normal subgroup of	C₂×#72232090, C₂×#72232175, C₂×#72232123, …

GAP

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

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

C2×#856585 #72325155