C₂×#7625730 #72298752

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7625730

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#3484	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

125125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72297215
maximal normal subgroup of	C₂×#72297215
maximal quotient of	C₂×#72263798

GAP

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

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

C2×#7625730 #72298752