#394473

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1955 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)(7,15,18,28)(8,12)(9,13)(10,14)(11,19,16,22)(17,33,34,31)(20,26)(21,27)(23,29)(24,30)(25,35,32,36);(5,6)(7,16,18,11)(8,12)(9,13)(10,14)(15,19,28,22)(17,32,34,25)(20,26)(21,27)(23,29)(24,30)(31,36,33,35);(5,7)(6,11)(8,17)(9,22)(10,18)(12,25)(13,28)(14,16)(15,30)(19,24)(20,36)(21,34)(23,35)(26,31)(27,32)(29,33);(2,4)(5,8,10,21)(6,12,14,27)(7,17,18,34)(9,23,24,20)(11,25,16,32)(13,29,30,26)(15,31,28,33)(19,36,22,35)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,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, 301669);