#142981

order	512
nilpotency class	4
isoclinism	isoclinic class of #142966
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	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

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

normal subgroups only

This group has 537 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278002
maximal normal subgroup of	#72278002

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50177);