#391843

order	512
nilpotency class	3
isoclinism	isoclinic class of #213744
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

978 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299039);