#110666

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,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, 17862);