#163779

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

708 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(18,22,24,27)(23,29,30,32)(33,34,37,42,38,41,36,40)(35,39,43,46,44,47,45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,23)(20,27)(21,24)(22,28)(25,30)(26,32)(29,31)(33,35)(34,39)(36,45)(37,43)(38,44)(40,48)(41,47)(42,46);(17,19,21,25)(18,23,24,30)(20,26,28,31)(22,29,27,32)(33,36,38,37)(34,40,41,42)(35,43,44,45)(39,46,47,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,25)(23,30)(26,31)(29,32)(34,41)(39,47)(40,42)(46,48)

Polycyclic presentation:

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

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, 70975);