#463157

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370353);