#376486

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283682);