#481784

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

993 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,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, 388980);