#318000

order	512
nilpotency class	3
isoclinism	isoclinic class of #317982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
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

498 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)(3,11)(4,9)(5,8)(6,10)(7,13)(12,15)(14,16)(17,18,22,28)(19,29,32,25)(20,33,35,39)(21,24,23,27)(26,36,42,34)(30,41,37,38)(31,45,46,47)(40,44,43,48);(1,3,6,14,4,13,5,12)(2,7,10,15,9,11,8,16)(18,24,28,27)(19,30,32,37)(25,29)(26,39,42,33)(31,44,46,48)(38,41)(40,43)(45,47);(2,8)(3,12)(5,6)(7,11)(9,10)(13,14)(17,19)(18,25)(20,31)(21,30)(22,32)(23,37)(24,38)(26,40)(27,41)(28,29)(33,47)(34,44)(35,46)(36,48)(39,45)(42,43);(2,9)(3,13)(8,10)(12,14)(17,20,21,34,22,35,23,36)(18,26,27,33,28,42,24,39)(19,31,30,44,32,46,37,48)(25,40,41,47,29,43,38,45)

Polycyclic presentation:

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

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, 225196);