#142983

order	512
nilpotency class	4
isoclinism	isoclinic class of #142969
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#3490	C₂²
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 647 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,h^b=hi,d^c=dh,f^c=fgi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231834
maximal normal subgroup of	#72231834

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50179);