#123588

order	512
nilpotency class	3
isoclinism	isoclinic class of #9103
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30784);