#237509

order	512
nilpotency class	3
isoclinism	isoclinic class of #237505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60803	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1327 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,e^c=ei,f^c=fhi,g^c=gi

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, 144705);