#481658

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

931 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,b^a=be,c^a=cg,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,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, 388854);