#132931

order	512
nilpotency class	4
isoclinism	isoclinic class of #132559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#52397	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

812 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40127);