#6668405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60375	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2548 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,24)(13,19)(14,22)(15,20)(16,21)(18,26)(23,33)(25,39)(27,31)(28,35)(29,37)(30,38)(32,36)(34,40);(1,3,4,8)(2,5,6,7)(9,11,16,24)(10,17,21,12)(13,28,30,34)(14,18,15,31)(19,35,38,40)(20,27,22,26)(23,32,39,29)(25,37,33,36);(9,12,16,17)(10,18,21,31)(11,22,24,20)(13,25,30,33)(14,27,15,26)(19,34,38,28)(23,37,39,36)(29,35,32,40);(2,6)(5,7);(3,8)(5,7)(9,13,16,30)(10,19,21,38)(11,23,24,39)(12,25,17,33)(14,29,15,32)(18,34,31,28)(20,36,22,37)(26,40,27,35)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,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,g^c=gi,h^c=hi,h^e=hi

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, 6575601);