#152959

order	512
nilpotency class	5
isoclinism	isoclinic class of #152957
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13225
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eh,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fgh,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60155);