#502193

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

760 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409389);