#105651

order	512
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 12847);