#501137

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#24416	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1116 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408333);