#483163

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

779 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)(3,4)(7,8)(9,10,13,20,12,26,27,35,14,17,29,19,15,18,30,36)(11,21,24,33,31,38,28,32,25,39,23,34,22,37,40,16);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,28)(14,25)(15,31)(17,32)(18,33)(19,37)(20,38)(21,35)(23,30)(24,27)(26,34)(29,40)(36,39);(1,3,5,4)(2,6,7,8)(10,17)(16,32)(18,26)(19,20)(21,39)(33,34)(35,36)(37,38);(9,12,14,15)(10,18,17,26)(11,22,25,31)(13,27,29,30)(16,33,32,34)(19,35,20,36)(21,38,39,37)(23,28,24,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309622, #72309653, #72309656, …
maximal normal subgroup of	#72309622, #72309653, #72309656, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390359);