#300526

order	512
nilpotency class	3
isoclinism	isoclinic class of #300501
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	#62558	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

888 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,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, 207722);