#204015

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	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=hi,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, 111211);