#142591

order	512
nilpotency class	4
isoclinism	isoclinic class of #142540
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

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

normal subgroups only

This group has 592 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,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 subgroup of	#72278278
maximal normal subgroup of	#72278278

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49787);