#139762

order	512
nilpotency class	4
isoclinism	isoclinic class of #139759
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#3486	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

656 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,f^d=fi,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, 46958);