#501074

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
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	#60458	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1088 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408270);