#6437558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6019 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344754);