#497181

order	512
nilpotency class	3
isoclinism	isoclinic class of #380391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2331
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22311
prime radical	C₂³	#568
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 176 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,f^b=fi,g^b=gi,g^c=gi

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, 404377);