#132074

order	512
nilpotency class	4
isoclinism	isoclinic class of #132074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8459
commutator	#117	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	#62373	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39270);