#495834

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

694 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403030);