#401008

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1656 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 308204);