#280914

order	512
nilpotency class	3
isoclinism	isoclinic class of #280883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

446 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=ehi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188110);