#420701

order	512
nilpotency class	3
isoclinism	isoclinic class of #417696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1162 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:

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

Polycyclic presentation:

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

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, 327897);