#478330

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

797 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)(5,6,9,15,10,14,22,13)(7,16,18,11,20,27,19,28)(8,21,24,12,25,31,23,30)(17,26,34,36,35,29,33,32);(5,7,9,18,10,20,22,19)(6,11,13,16,14,28,15,27)(8,17,24,34,25,35,23,33)(12,29,21,36,30,26,31,32);(5,8,22,23,10,25,9,24)(6,12,15,31,14,30,13,21)(7,17,19,33,20,35,18,34)(11,26,27,36,28,29,16,32);(2,4)(6,13)(8,23)(9,22)(11,16)(14,15)(17,33)(18,19)(21,31)(24,25)(27,28)(32,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385526);