#149860

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	#11530
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2960 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 252 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dhi,e^b=efh,f^b=fi,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, 57056);