#445646

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

644 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)(9,10,16,22)(11,18,25,33)(12,17,24,34)(13,27)(14,21,15,20)(19,29)(23,39)(26,35)(28,36)(30,38)(31,37)(32,40);(1,3,5,8)(2,6,4,7)(9,11,16,25)(10,17,22,34)(12,14,24,15)(13,28,31,26)(18,20,33,21)(19,36,38,35)(23,30,40,29)(27,32,37,39);(9,12,14,25,16,24,15,11)(10,18,20,34,22,33,21,17)(13,26,30,40,31,28,29,23)(19,35,37,39,38,36,27,32);(1,4,5,2)(3,6,8,7)(9,13)(10,19)(11,23)(12,26)(14,30)(15,29)(16,31)(17,32)(18,35)(20,37)(21,27)(22,38)(24,28)(25,40)(33,36)(34,39)

Polycyclic presentation:

2^9|a^2=eg,b^2=efi,c^2=g,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,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, 352842);