#420930

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 328126);