#163233

order	512
nilpotency class	3
isoclinism	isoclinic class of #162662
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

772 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)(4,8)(5,7)(10,15,16,13)(12,14)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32)(33,34,37,42,38,40,36,41)(35,43,46,39,44,48,45,47);(1,3,5,11)(2,6,7,9)(4,10,12,16)(8,13,14,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26)(33,35)(34,39)(36,46)(37,45)(38,44)(40,47)(41,48)(42,43);(2,7)(6,9)(8,14)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(34,40)(35,44)(41,42)(45,46);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(34,40)(39,47)(41,42)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70429);