#418154

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 325350);