#387071

order	512
nilpotency class	3
isoclinism	isoclinic class of #213752
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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1052 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)(3,5)(4,6)(7,8)(9,10,14,20,15,19,13,18)(11,21,23,31,24,16,28,32)(12,25,26,35,27,17,29,34)(22,33,38,40,37,30,36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,33)(18,32)(19,21)(20,31)(25,30)(26,36)(27,37)(29,38)(34,39)(35,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,22)(13,26)(14,29)(15,27)(16,30)(18,34)(19,25)(20,35)(21,33)(23,36)(24,37)(28,38)(31,39)(32,40);(2,3)(6,7)(9,13,15,14)(10,18,19,20)(11,23,24,28)(12,26,27,29)(16,31,21,32)(17,34,25,35)(22,36,37,38)(30,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294267);