#163264

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

779 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)(4,12)(5,7)(8,13)(10,14,16,15)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30)(33,34,37,42,38,40,36,41)(35,43,46,39,44,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29)(33,35)(34,39)(36,46)(37,45)(38,44)(40,47)(41,48)(42,43);(2,7)(6,9)(8,12)(14,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(34,40)(35,44)(41,42)(45,46);(1,4,5,13)(2,8,7,12)(3,10,11,16)(6,14,9,15)(34,40)(39,47)(41,42)(43,48)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=g,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, 70460);