#273609

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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)(3,10,11,15)(4,6,13,8)(7,14,16,12)(17,18)(19,27,31,24)(20,22)(21,32,36,25)(23,26)(28,30,42,38)(29,39)(33,37)(34,40)(35,45,48,41)(43,44)(46,47);(2,6)(3,11)(4,13)(7,15)(8,9)(10,16)(17,19)(18,24)(20,27)(21,34)(22,31)(23,38)(25,40)(26,42)(28,33)(29,36)(30,37)(32,39)(35,44)(41,43)(45,46)(47,48);(17,20,22,18)(19,28,31,42)(21,32,36,25)(23,26,37,33)(24,38,27,30)(29,43,34,46)(35,41,48,45)(39,47,40,44);(1,3,5,11)(2,7,9,16)(4,14,13,12)(6,10,8,15)(17,21)(18,25)(19,29)(20,32)(22,36)(23,35)(24,39)(26,41)(27,40)(28,43)(30,44)(31,34)(33,45)(37,48)(38,47)(42,46)

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180805);