#198323

order	512
nilpotency class	3
isoclinism	isoclinic class of #198144
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105519);