#450127

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1065 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,d^2=e,f^2=i,g^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,h^b=hi,f^c=fi,g^c=gi,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, 357323);