#446202

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

714 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)(7,17,21,29)(8,22,26,30)(9,15)(10,13)(11,16)(12,20,27,18)(14,25,32,24)(19,28)(23,31)(33,34)(35,36);(5,7)(6,12,16,27)(8,23,26,34)(9,20)(10,18)(11,21)(13,29,15,17)(14,31)(19,24,35,25)(22,28)(30,36)(32,33);(6,13,16,15)(7,18,21,20)(12,27)(14,30,32,22)(17,29)(19,23,35,34)(28,36)(31,33);(2,4)(5,8,10,24,11,26,9,25)(6,14,15,22,16,32,13,30)(7,19,18,23,21,35,20,34)(12,28,17,33,27,36,29,31)

Polycyclic presentation:

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

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, 353398);