#274040

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

863 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=h,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181236);