#416869

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

638 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)(5,6,11,15,9,14,10,24)(7,12,19,27,17,29,18,28)(8,20,25,31,22,36,23,13)(16,32,34,26,21,35,33,30);(5,7,11,18,9,17,10,19)(6,12,15,28,14,29,24,27)(8,16,25,33,22,21,23,34)(13,26,20,32,31,30,36,35);(2,4)(5,7,9,17)(6,12)(8,21,22,16)(10,19,11,18)(13,30)(14,29)(15,28)(20,35)(23,33,25,34)(24,27)(26,31)(32,36);(5,8,10,23,9,22,11,25)(6,13,15,20,14,31,24,36)(7,16,19,34,17,21,18,33)(12,26,28,32,29,30,27,35)

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=e,d^2=h,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,f^b=fi,g^b=gi,d^c=di,f^c=fi,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, 324065);