#434712

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1112 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,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, 341908);