#164806

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=g,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dgi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72002);