#498320

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×#2213	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

646 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,d^b=de,e^b=ei,g^b=gi,f^c=fh,h^c=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, 405516);