#418762

order	512
nilpotency class	3
isoclinism	isoclinic class of #417489
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

1246 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=e,f^2=i,g^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,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, 325958);