#449594

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60367	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2054 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)(3,7,8,5)(9,10,14,19,13,17,15,20)(11,21,22,31,24,16,25,35)(12,26,30,18,28,36,29,33)(23,38,39,34,27,40,37,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,27)(13,24)(14,22)(15,25)(17,21)(18,34)(19,35)(20,31)(23,28)(26,38)(29,39)(30,37)(32,33)(36,40);(10,17)(11,22)(14,15)(16,31)(18,33)(21,35)(23,37)(24,25)(27,39)(29,30)(32,38)(34,40);(3,8)(5,7)(9,12)(10,18)(11,23)(13,28)(14,29)(15,30)(16,32)(17,33)(19,26)(20,36)(21,34)(22,37)(24,27)(25,39)(31,38)(35,40)

Polycyclic presentation:

2^9|a^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=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, 356790);