#394467

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
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	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2000 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)(5,6,9,12)(7,16)(8,14,21,26)(10,15,11,24)(13,17)(18,33)(19,25)(20,27)(22,29,23,30)(28,35)(31,34)(32,36);(6,12)(7,17,20,19)(13,27,25,16)(14,26)(15,24)(18,32,34,35)(28,33,36,31)(29,30);(5,7)(6,13)(8,18)(9,20)(10,17)(11,19)(12,25)(14,28)(15,27)(16,24)(21,34)(22,35)(23,32)(26,36)(29,31)(30,33);(2,4)(5,8)(6,14)(7,18)(9,21)(10,23)(11,22)(12,26)(13,28)(15,30)(16,31)(17,32)(19,35)(20,34)(24,29)(25,36)(27,33)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^c=ei,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, 301663);