#403976

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62855	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2627 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311172);