#468415

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

666 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:

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

Polycyclic presentation:

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

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, 375611);