#501078

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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

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:

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

Polycyclic presentation:

2^9|a^2=fi,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408274);