#411829

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

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

2044 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)(5,6)(7,8)(9,10,16,22)(11,19,25,32)(12,18,29,37)(13,17,24,33)(14,21,15,20)(23,35,39,34)(26,40,27,31)(28,38,30,36);(1,2,3,4)(9,11,14,13,16,25,15,24)(10,17,20,19,22,33,21,32)(12,26,28,39,29,27,30,23)(18,34,36,40,37,35,38,31);(6,8)(9,12)(10,18)(11,23)(13,27)(14,30)(15,28)(16,29)(17,31)(19,35)(20,38)(21,36)(22,37)(24,26)(25,39)(32,34)(33,40);(9,13,14,25,16,24,15,11)(10,19,20,33,22,32,21,17)(12,27,30,39,29,26,28,23)(18,35,38,40,37,34,36,31)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=hi,e^2=i,g^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=di,e^c=ei,g^c=gi

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, 319025);