#117497

order	512
nilpotency class	3
isoclinism	isoclinic class of #117489
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

584 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)(3,11,14,16,12,15,13,7)(17,18)(19,28)(20,33,37,25)(21,38)(22,26)(23,27)(24,30)(29,44,47,41)(31,45)(32,40)(34,39,46,48)(35,42,36,43);(1,3,4,13,6,12,5,14)(2,7,8,11,10,16,9,15)(17,19,21,30,23,31,22,32)(18,24,26,28,27,40,38,45)(20,34,35,29,37,46,36,47)(25,41,42,48,33,44,43,39);(2,8)(3,12)(4,5)(7,15)(9,10)(11,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,37)(24,39)(26,43)(27,33)(28,44)(30,34)(31,47)(32,46)(38,42)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=ehi,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24693);