#201273

order	512
nilpotency class	3
isoclinism	isoclinic class of #201121
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37616	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108469);