#6669059

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61055	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,19)(13,18)(14,17)(15,20)(16,24)(22,32)(23,37)(25,31)(26,40)(27,34)(28,33)(29,36)(30,39)(35,38);(2,5)(3,7)(9,11)(10,16)(12,26)(13,24)(14,25)(15,22)(17,32)(18,21)(19,35)(20,31)(23,29)(27,38)(28,39)(30,36)(33,37)(34,40);(1,3,4,7)(2,6,5,8)(10,17)(11,22)(12,27)(14,15)(18,20)(19,36)(23,26)(24,25)(30,37)(31,32)(33,34)(38,39);(2,5)(6,8)(10,18)(16,21)(17,20)(19,34)(30,37)(31,32)(33,36)(35,40);(9,12,13,27)(10,19,18,34)(11,23,24,39)(14,28,15,29)(16,30,21,37)(17,33,20,36)(22,38,25,26)(31,40,32,35)

Polycyclic presentation:

2^9|c^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576255);