#206434

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	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1044 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=dhi

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, 113630);