#363269

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270465);