#299372

order	512
nilpotency class	3
isoclinism	isoclinic class of #299341
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 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:

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

Polycyclic presentation:

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

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, 206568);