#482143

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

860 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cgh,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, 389339);