#473993

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,g^d=gi,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, 381189);