#363226

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2464 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270422);