#163287

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

760 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)(3,11,14,7)(4,9)(5,8)(6,10)(12,15,13,16)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30)(33,34,37,42,38,40,36,41)(35,43,46,39,44,48,45,47);(1,3,6,14)(2,7,10,11)(4,12,5,13)(8,15,9,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32)(33,35)(34,39)(36,46)(37,45)(38,44)(40,47)(41,48)(42,43);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(18,24)(23,31)(25,26)(27,32)(34,40)(35,44)(41,42)(45,46);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)(34,40)(39,47)(41,42)(43,48)

Polycyclic presentation:

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

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, 70483);