#394516

order	512
nilpotency class	3
isoclinism	isoclinic class of #393975
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	#35149	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2307 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 301712);