#448890

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

838 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)(5,6,10,26,11,16,22,15)(7,17,20,12,21,33,34,28)(8,13,25,30,23,31,9,14)(18,32,24,29,19,36,35,27);(5,7)(6,12,16,28)(8,19,23,18)(9,24,25,35)(10,20)(11,21)(13,29)(14,32)(15,17,26,33)(22,34)(27,31)(30,36);(5,8,22,25,11,23,10,9)(6,13,26,14,16,31,15,30)(7,18,34,24,21,19,20,35)(12,27,33,36,28,29,17,32);(2,4)(5,9,10,23,11,25,22,8)(6,14,15,13,16,30,26,31)(7,19,34,35,21,18,20,24)(12,27,33,36,28,29,17,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=ei,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^c=hi,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, 356086);