#498772

order	512
nilpotency class	3
isoclinism	isoclinic class of #391902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23366
prime radical	C₂³	#568
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

332 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405968);