C₂×#7670725 #72313633

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7670725

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

118285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72311661, C₂×#72311691, #72314394
maximal normal subgroup of	C₂×#72311661, C₂×#72311691, #72314394

GAP

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

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

C2×#7670725 #72313633