#163312

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
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

706 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,25)(20,29)(21,26)(22,31)(23,27)(24,28)(30,32)(33,34,36,40,37,39,43,46)(35,38,42,45,41,44,47,48);(1,3,5,11)(2,6,7,14)(4,12,13,10)(8,16,15,9)(17,19,23,20)(18,24,27,32)(21,28,22,30)(25,26,29,31)(33,35)(34,38)(36,42)(37,41)(39,44)(40,45)(43,47)(46,48);(2,7)(6,14)(8,15)(9,16)(17,20,23,19)(18,25,27,29)(21,28,22,30)(24,26,32,31)(35,41)(38,44)(42,47)(45,48);(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,29,32,25)(34,39)(38,44)(40,46)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70508);