#269588

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,12,20,11,13,10,14)(7,15,18,17,9,8,16,19)(21,22)(23,28)(24,32)(25,27)(26,30)(29,34)(31,36)(33,35);(1,2)(3,4)(5,7,11,9)(6,8,13,15)(10,18,12,16)(14,19,20,17)(22,27)(23,25)(28,30)(29,35)(31,34)(32,36);(2,4)(5,8)(6,7)(9,13)(10,19)(11,15)(12,17)(14,18)(16,20)(21,23,26,25)(22,28,30,27)(24,31,33,34)(29,32,36,35);(5,9,11,7)(6,8,13,15)(10,16,12,18)(14,19,20,17)(21,24,26,33)(22,29,30,36)(23,31,25,34)(27,35,28,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=i,d^2=h,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=deh,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176784);