#435971

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

757 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,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, 343167);