#497893

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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)(5,6,12,18)(7,19,22,32)(8,16,26,35)(9,27,29,34)(10,31,25,14)(11,28,33,17)(13,20,30,15)(21,24,23,36);(5,7)(6,13)(8,15)(9,28)(10,27)(11,21)(12,22)(14,32)(16,24)(17,29)(18,30)(19,31)(20,26)(23,33)(25,34)(35,36);(5,8,11,25,12,26,33,10)(6,14,17,16,18,31,28,35)(7,20,21,27,22,15,23,34)(9,24,13,19,29,36,30,32);(5,9)(6,15)(7,16)(8,24)(10,32)(11,13)(12,29)(14,23)(17,34)(18,20)(19,25)(21,31)(22,35)(26,36)(27,28)(30,33)

Polycyclic presentation:

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

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, 405089);