#435972

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,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, 343168);