#114363

order	512
nilpotency class	3
isoclinism	isoclinic class of #114344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

894 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21559);