#198122

order	512
nilpotency class	3
isoclinism	isoclinic class of #198117
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	C₂×#3207	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1232 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105318);