#501626

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 ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

677 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 408822);