#508670

order	512
nilpotency class	4
isoclinism	isoclinic class of #508669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

485 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)(3,6)(4,5)(7,8)(9,10,16,22,17,12,14,20)(11,23,27,18,28,36,26,29)(13,30,32,21,33,19,15,31)(24,39,35,25,37,34,40,38);(2,5)(6,8)(9,11)(10,18)(12,29)(13,25)(14,27)(15,34)(16,26)(17,28)(19,35)(20,36)(21,37)(22,23)(24,31)(30,40)(32,39)(33,38);(1,3)(2,6)(4,7)(5,8)(9,12,17,10)(11,24,28,37)(13,31,33,21)(14,20,16,22)(15,30,32,19)(18,34,29,39)(23,38,36,25)(26,35,27,40);(3,7)(6,8)(9,13,14,15,17,33,16,32)(10,19,20,21,12,30,22,31)(11,25,27,34,28,38,26,39)(18,35,36,37,29,40,23,24)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=f,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,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, 415866);