#150843

order	512
nilpotency class	5
isoclinism	isoclinic class of #150842
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11878
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=ghi,d^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efgi,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58039);