#333267

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

577 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240463);