#106961

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,6,10,5,9)(3,11,12,15,13,16,14,7)(17,18,21,27,23,26,22,28)(19,29,32,39,33,24,31,40)(20,34,35,41,36,42,37,25)(30,44,46,43,47,48,45,38);(1,3,5,12,6,13,4,14)(2,7,9,11,10,15,8,16)(17,19,22,32,23,33,21,31)(18,24,28,40,26,29,27,39)(20,30,35,45,36,47,37,46)(25,38,34,48,41,43,42,44);(17,20,23,36)(18,25,26,41)(19,30,33,47)(21,37,22,35)(24,38,29,43)(27,42,28,34)(31,46,32,45)(39,44,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14157);