#145682

order	512
nilpotency class	4
isoclinism	isoclinic class of #145666
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=d,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,e^c=ei,f^c=fgi,h^c=hi,f^e=fi

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, 52878);