#395787

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1827 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:

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

Polycyclic presentation:

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

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, 302983);