#481690

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1015 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,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, 388886);