#404632

order	512
nilpotency class	3
isoclinism	isoclinic class of #404632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

733 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)(6,11)(7,16)(8,21)(9,23)(12,26)(13,28)(17,24)(19,20)(22,34)(25,29)(30,32)(31,36);(5,6,9,15,10,11,23,14)(7,17,16,12,20,26,19,24)(8,13,21,29,22,25,34,28)(18,31,30,27,33,36,32,35);(5,7)(6,12,11,24)(8,18)(9,19)(10,20)(13,27,25,35)(14,26,15,17)(16,23)(21,32)(22,33)(28,36,29,31)(30,34);(2,4)(5,8,9,21,10,22,23,34)(6,13,15,29,11,25,14,28)(7,18,19,32,20,33,16,30)(12,27,17,31,24,35,26,36)

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=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, 311828);