#424929

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,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, 332125);