#163907

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1146 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,e^2=h,f^2=i,b^a=be,c^a=ci,e^a=eg,e^b=eh,f^b=fgh,d^c=dh

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, 71103);