#203925

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=di,e^b=eh

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, 111121);