#482137

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

907 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)(11,21,25,29)(12,19,14,17)(13,18)(15,20)(16,24)(22,31)(23,35,38,39)(26,33,27,32)(28,34)(30,37)(36,40);(1,2,4,6)(3,8,7,5)(9,11)(10,16,20,31)(12,22)(13,23)(14,24)(15,25)(17,29,19,21)(18,30,34,40)(26,36)(27,37)(28,38)(32,39,33,35);(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,27,28,26)(16,29,31,21)(18,33,34,32)(23,37,38,36)(30,35,40,39);(9,13,15,28)(10,18,20,34)(11,23,25,38)(12,26,14,27)(16,30,31,40)(17,32,19,33)(21,35,29,39)(22,36,24,37)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,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, 389333);