C₂×C₅²×#509 #72253265

order	3200
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅² · Schur multiplier (H₂) = C₂⁵×C₅

Aliases

direct product

C₂×C₅²×#509

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅²	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₅²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₂²×C₄×C₅² ◁ C₂²×C₅²×C₈ ◁ G

1600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 4 generators on 28 points:

(3,4,5,10,13,17,18,15,7,9,14,16,6,11,12,8)(19,20,21,22,23);(1,2)(24,25,26,27,28);(4,8)(5,12)(6,13)(9,15)(10,11)(14,18)(16,17);(4,9)(8,15)(10,16)(11,17)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×C₅⋊D₅×#509, C₂×#509×#1043
maximal subgroup of	C₂×C₅⋊D₅×#509
maximal normal subgroup of	C₂×C₅⋊D₅×#509

GAP

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

gap> G := Group((3,4,5,10,13,17,18,15,7,9,14,16,6,11,12,8)(19,20,21,22,23), (1,2)(24,25,26,27,28), (4,8)(5,12)(6,13)(9,15)(10,11)(14,18)(16,17), (4,9)(8,15)(10,16)(11,17));

C2×C52×#509 #72253265