#501251

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61056	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408447);