#142573

order	512
nilpotency class	4
isoclinism	isoclinic class of #142540
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×C₄⋊C₄	C₂³
prime radical	#3492	C₂²
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

656 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227650

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49769);