#204019

order	512
nilpotency class	3
isoclinism	isoclinic class of #19304
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

545 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)(17,18,22,24)(19,26,29,32)(20,30,25,21)(23,28,31,27);(1,3,5,11,6,12,4,14)(2,7,9,16,8,15,13,10)(17,19,22,29)(18,23,24,31)(20,26,25,32)(21,28,30,27);(17,20,21,18,22,25,30,24)(19,26,28,23,29,32,27,31);(2,8)(7,15)(9,13)(10,16)(18,24)(20,25)(23,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111215);