#149819

order	512
nilpotency class	5
isoclinism	isoclinic class of #149794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11513
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₂×C₄² ◁ #2920 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,e^b=efhi,e^c=ei,e^d=ehi,f^e=fi

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, 57015);