#450927

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1188 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)(9,10,13,19,14,18,28,17)(11,20,22,30,23,15,27,31)(12,24,26,33,25,16,32,34)(21,29,37,40,36,35,38,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,23)(10,15,18,20)(12,21,25,36)(13,27,28,22)(16,29,24,35)(17,31,19,30)(26,38,32,37)(33,40,34,39);(3,7)(5,8)(9,12)(10,16,18,24)(11,21)(13,26)(14,25)(15,29,20,35)(17,33,19,34)(22,37)(23,36)(27,38)(28,32)(30,39,31,40);(10,17)(13,28)(15,30)(16,33)(18,19)(20,31)(22,27)(24,34)(26,32)(29,39)(35,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358123);