#476528

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383724);