#265370

order	512
nilpotency class	3
isoclinism	isoclinic class of #265359
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#525	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

832 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18,23,28)(19,25,31,40)(20,32,36,47)(21,37,22,27)(24,38,41,30)(26,35,44,34)(29,45,33,39)(42,48,43,46);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,22,38,23,31,21,30)(18,24,37,40,28,41,27,25)(20,29,35,48,36,33,34,46)(26,39,47,43,44,45,32,42);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,19)(18,25)(20,33)(21,30)(22,38)(23,31)(24,37)(26,43)(27,41)(28,40)(29,36)(32,39)(34,48)(35,46)(42,44)(45,47);(17,20,22,35,23,36,21,34)(18,26,37,47,28,44,27,32)(19,29,38,48,31,33,30,46)(24,39,40,43,41,45,25,42)

Polycyclic presentation:

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

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, 172566);