#107834

order	512
nilpotency class	3
isoclinism	isoclinic class of #107810
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

781 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15030);