#456092

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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)(11,15)(12,21)(13,27)(14,19)(16,20)(17,25)(18,31)(22,29)(23,37)(24,34)(26,33)(28,40)(30,36)(32,38)(35,39);(2,5)(6,8)(9,11,15,10,16,19,14,20)(12,21,25,34,26,33,24,17)(13,22,30,18,31,36,29,27)(23,37,38,40,39,35,28,32);(3,7)(6,8)(9,12)(10,17)(11,21)(13,28)(14,25)(15,24)(16,26)(18,35)(19,33)(20,34)(22,32)(23,29)(27,37)(30,39)(31,38)(36,40);(1,3,4,7)(2,6,5,8)(9,13,14,29,16,31,15,30)(10,18,11,22,20,27,19,36)(12,23,24,28,26,39,25,38)(17,32,33,35,34,40,21,37)

Polycyclic presentation:

2^9|b^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 363288);