#430121

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

936 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^d=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, 337317);