#438561

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

825 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=efi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi,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, 345757);