C₂×#856561 #72324051

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#856561

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55189 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)(19,20)(21,22)(23,24)(25,26);(7,8)(9,10)(11,12,15,14)(13,16,17,18)(19,20,23,24)(21,22,25,26);(4,6)(20,24)(22,26);(1,2);(4,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,22)(23,25)(24,26);(3,4)(5,6)(11,14,15,12)(13,16,17,18)(19,22)(20,25)(21,24)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232332, C₂×#72323256, C₂×#72323274
maximal normal subgroup of	C₂×#72232332, C₂×#72323256, C₂×#72323274

GAP

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

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

C2×#856561 #72324051