#107493

order	512
nilpotency class	3
isoclinism	isoclinic class of #107474
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

699 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14689);