#106457

order	512
nilpotency class	3
isoclinism	isoclinic class of #106434
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

762 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:

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

Polycyclic presentation:

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

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, 13653);