#134913

order	512
nilpotency class	4
isoclinism	isoclinic class of #134913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1799
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2388	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

387 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=gi,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fi,e^c=egi,f^c=fghi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42109);