#479813

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

607 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,f^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,g^b=gi,d^c=de,e^c=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387009);