C₂×#19083 #6351497

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#19083

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2852 conjugacy classes of subgroups; only the 594 normal subgroups are shown.

normal subgroups only

This group has 2852 conjugacy classes of subgroups — too many to draw. Only its 594 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(4,9)(5,13)(6,7)(11,12)(14,15)(16,17)(20,25)(22,23)(26,33)(27,31)(28,29)(32,34);(3,4)(5,10)(6,11)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20,24,27)(21,26,30,34)(22,25,23,31)(28,33,29,32);(3,5,8,15)(4,10,12,18)(6,13,7,14)(9,16,11,17)(19,21,24,30)(20,26,27,34)(22,29,23,28)(25,32,31,33);(3,6,8,7)(4,11,12,9)(5,14,15,13)(10,16,18,17)(19,22,24,23)(20,25,27,31)(21,28,30,29)(26,33,34,32)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258693);