#430457

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1223 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:

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

Polycyclic presentation:

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

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, 337653);