#106217

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,8,11,9,7,12,10)(13,14)(15,20,25,35)(16,27,28,38)(17,23)(18,32)(19,22)(21,29,37,31)(24,39)(26,34,30,36)(33,41)(40,44)(42,43);(2,4)(5,7)(6,9)(8,10)(11,12)(13,15,18,26,19,25,17,30)(14,20,23,36,22,35,32,34)(16,24,29,41,28,42,31,40)(21,33,38,39,37,44,27,43);(5,6)(7,9)(8,11)(10,12)(13,16)(14,21)(15,24)(17,31)(18,29)(19,28)(20,33)(22,37)(23,38)(25,42)(26,41)(27,32)(30,40)(34,43)(35,44)(36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=dh,e^b=eh,f^b=fhi,d^c=dh,e^c=eh,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13413);