#474874

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

526 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,21,23,32)(12,26)(13,19,14,18)(16,25,33,24)(17,28)(22,37)(27,31)(29,34)(30,35)(36,39)(38,40);(2,5)(6,7)(9,11,13,24,15,23,14,25)(10,16,18,21,20,33,19,32)(12,27,28,38,30,22,29,39)(17,31,26,36,34,37,35,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,30)(10,17)(11,22,23,27)(13,28,14,29)(16,31)(18,26)(19,35)(20,34)(21,36)(24,39,25,38)(32,40)(33,37);(3,8)(6,7)(10,18,20,19)(11,23)(12,28,30,29)(16,32,33,21)(17,34)(22,38,27,39)(24,25)(26,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=efgh,c^2=g,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=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, 382070);