#93661

order	512
nilpotency class	4
isoclinism	isoclinic class of #93616
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1309
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂	#7050
prime radical	C₂³	#338
LCS	G ▷ C₂²×C₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=h,e^2=fgh,h^2=i,b^a=bc,c^a=cf,e^a=efghi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,h^b=hi

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, 857);