#117689

order	512
nilpotency class	3
isoclinism	isoclinic class of #117681
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=ei,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24885);