#108477

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

422 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:

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

Polycyclic presentation:

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

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, 15673);