#147135

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#387
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1917
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=di,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54331);