#142532

order	512
nilpotency class	4
isoclinism	isoclinic class of #142484
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	#537	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

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

normal subgroups only

This group has 457 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,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,f^b=fi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,f^d=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, 49728);