#436612

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343808);