#431343

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=f,d^2=efi,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,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, 338539);