#501247

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
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

1083 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:

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

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,e^b=eh,h^b=hi,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408443);