#418012

order	512
nilpotency class	3
isoclinism	isoclinic class of #417468
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi

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, 325208);