#206461

order	512
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

922 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci,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, 113657);