#501072

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

1101 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 408268);