#501076

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1096 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 408272);