#267870

order	512
nilpotency class	3
isoclinism	isoclinic class of #267851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175066);