#464206

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1779 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371402);