#419841

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1574 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)(5,6,9,15,11,14,10,26)(7,16,18,31,20,12,19,28)(8,21,23,30,25,32,24,13)(17,34,35,27,36,29,22,33);(5,7)(6,12)(8,22)(9,19)(10,18)(11,20)(13,29)(14,16)(15,31)(17,24)(21,27)(23,36)(25,35)(26,28)(30,34)(32,33);(2,4)(5,8,11,25)(6,13,14,30)(7,17,20,36)(9,24,10,23)(12,27,16,33)(15,32,26,21)(18,22,19,35)(28,34,31,29);(5,7)(6,12,14,16)(8,17)(9,18)(10,19)(11,20)(13,27,30,33)(15,28,26,31)(21,29,32,34)(22,24)(23,35)(25,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327037);