#200108

order	512
nilpotency class	3
isoclinism	isoclinic class of #199851
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	C₂³×C₄	C₂⁴
prime radical	#60805	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1220 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)(3,6,10,15)(4,11)(5,8)(7,13)(9,16,12,14)(17,18,21,26,22,24,20,25)(19,27,29,23,30,32,28,31);(1,3)(2,6,8,15)(4,12)(5,10)(7,16,11,14)(9,13)(17,19)(18,23)(20,29)(21,28)(22,30)(24,31)(25,32)(26,27);(4,13)(7,11)(9,12)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,4,5,13)(2,7,8,11)(3,9,10,12)(6,14,15,16)(18,24)(23,31)(25,26)(27,32)

Polycyclic presentation:

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

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, 107304);