C₂×#556176 #72331940

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#556176

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

64709 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)(12,15)(13,17)(19,20)(21,25)(22,23)(24,26);(3,4)(5,6)(7,8)(9,10)(11,12,14,15)(13,16,17,18)(19,21,23,26)(20,24,22,25);(19,21)(20,25)(22,24)(23,26);(1,2);(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,25)(23,26);(11,13)(12,16)(14,17)(15,18)(19,22,23,20)(21,24,26,25)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323302, C₂×#72323316, C₂×#72323326
maximal normal subgroup of	C₂×#72323302, C₂×#72323316, C₂×#72323326

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

C2×#556176 #72331940