#204012

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 111208);