#6437562

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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^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,e^b=ei,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, 6344758);