#146204

order	512
nilpotency class	4
isoclinism	isoclinic class of #146204
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

245 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=de,d^2=g,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53400);