#479821

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

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

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, 387017);