#161719

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

788 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)(3,10,13,7)(4,14)(5,8)(6,9)(11,15,12,16)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31)(33,34,37,41,38,42,36,40)(35,39,45,48,43,46,44,47);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,23)(22,30)(24,31)(25,32)(35,43)(39,46)(44,45)(47,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(33,36,38,37)(34,40,42,41)(35,44,43,45)(39,47,46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cgh,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68915);