C₂×#8533 #187897

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#8533

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2094 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 2094 conjugacy classes of subgroups — too many to draw. Only its 338 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 — 4 generators on 34 points:

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

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, 95093);