#6814833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
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	#60768	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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)(11,20)(12,16)(13,18)(14,19)(15,24)(17,29)(21,22)(23,35)(25,28)(26,36)(27,33)(30,37)(31,38)(32,39)(34,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,21)(12,25)(13,17)(14,19)(15,24)(16,28)(18,29)(20,22)(23,32)(26,31)(27,34)(30,37)(33,40)(35,39)(36,38);(9,11,14,22)(10,15)(12,26,27,23)(13,24,29,37)(16,32)(17,20)(18,21)(19,30)(25,35)(28,39,40,38)(31,33)(34,36);(2,6)(5,7)(11,22)(12,27)(15,30)(16,33)(20,21)(24,37)(25,34)(28,40);(3,8)(5,7)(9,12,13,28,14,27,29,40)(10,16,18,25,19,33,17,34)(11,23,24,38,22,26,37,39)(15,31,21,36,30,32,20,35)

Polycyclic presentation:

2^9|c^2=gh,e^2=h,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,d^c=di,e^c=ei,e^d=ei

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, 6722029);