#219664

order	512
nilpotency class	3
isoclinism	isoclinic class of #219576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ei,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,g^c=gh

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, 126860);