#424738

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,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, 331934);