#108023

order	512
nilpotency class	3
isoclinism	isoclinic class of #108006
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

817 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:

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

Polycyclic presentation:

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

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, 15219);