#403072

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

851 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)(5,6,11,16,9,17,10,15)(7,12,20,28,19,29,26,13)(8,21,25,33,23,32,24,14)(18,30,34,27,22,31,35,36);(5,7,10,20,9,19,11,26)(6,12,15,28,17,29,16,13)(8,18,24,34,23,22,25,35)(14,27,32,31,33,36,21,30);(5,7,9,19)(6,13)(8,22,23,18)(10,20,11,26)(12,15)(14,31)(16,29)(17,28)(21,27)(24,35,25,34)(30,33)(32,36);(2,4)(5,8,11,25,9,23,10,24)(6,14,15,32,17,33,16,21)(7,18,26,35,19,22,20,34)(12,27,28,31,29,36,13,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310268);