#429471

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1198 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=efi,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,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 336667);