#471079

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 378275);