#417413

order	512
nilpotency class	3
isoclinism	isoclinic class of #416736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2862
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15253
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 749 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=f,d^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324609);