#142950

order	512
nilpotency class	4
isoclinism	isoclinic class of #142933
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	#18337	C₂
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

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

normal subgroups only

This group has 485 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,c^b=cf,d^b=dg,f^b=fg,h^b=hi,d^c=dh,e^c=egi,f^c=fg,f^d=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, 50146);