#391653

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 298849);