#445120

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

985 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)(3,6)(4,5)(7,8)(9,10,14,16,15,20,13,19)(11,17,21,29,24,30,23,32)(12,25,26,33,28,34,27,18)(22,36,35,39,37,40,38,31);(10,16)(11,21)(12,26)(13,14)(17,30)(18,33)(19,20)(22,37)(23,24)(27,28)(31,40)(36,39);(2,5)(6,8)(9,11,14,21,15,24,13,23)(10,17,16,29,20,30,19,32)(12,22,26,35,28,37,27,38)(18,31,25,36,33,39,34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,27)(14,26)(15,28)(16,25)(17,31)(19,34)(20,33)(21,35)(23,38)(24,37)(29,36)(30,39)(32,40)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^d=ei,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, 352316);