#501627

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
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

645 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ei,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=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, 408823);