#500111

order	512
nilpotency class	4
isoclinism	isoclinic class of #500109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407307);