#423571

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330767);