#394663

order	512
nilpotency class	3
isoclinism	isoclinic class of #394102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=e,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,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301859);