#479866

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efgh,c^2=efi,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,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 387062);