#285470

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18585	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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:

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

Polycyclic presentation:

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

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, 192666);