#417957

order	512
nilpotency class	3
isoclinism	isoclinic class of #417597
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2013 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,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, 325153);