#417430

order	512
nilpotency class	3
isoclinism	isoclinic class of #416472
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2864
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15255
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

573 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 573 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

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

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, 324626);