#137443

order	512
nilpotency class	4
isoclinism	isoclinic class of #137387
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

425 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=g,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eghi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264974, #72265509, #72265512
maximal normal subgroup of	#72264974, #72265509, #72265512

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44639);