#480300

order	512
nilpotency class	4
isoclinism	isoclinic class of #480250
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1038 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fg,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, 387496);