#495793

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3035	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,f^c=fi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402989);