#481621

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1106 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)(11,21,24,34)(12,19,14,17)(13,18)(15,20)(16,22)(23,35,38,39)(25,32,26,31)(27,33)(28,30)(29,36)(37,40);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,21)(18,29)(19,34)(20,30)(25,36)(26,37)(27,38)(31,35)(32,39)(33,40);(2,6)(5,7)(9,12,15,14)(10,17,20,19)(11,22,24,28)(13,26,27,25)(16,21,30,34)(18,32,33,31)(23,37,38,36)(29,39,40,35);(9,13,15,27)(10,18,20,33)(11,23,24,38)(12,25,14,26)(16,29,30,40)(17,31,19,32)(21,35,34,39)(22,36,28,37)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi

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, 388817);