#149305

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#387
commutator	C₂²×C₄	C₂²×C₈
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

478 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)(5,6,11,17,9,16,27,36,12,18,24,34,10,15,20,30)(7,19,8,23,21,13,25,32,22,31,26,35,28,33,29,14);(2,4)(5,7,12,22)(6,13)(8,24,26,11)(9,21,10,28)(14,34)(15,19)(16,31)(17,32)(18,33)(20,29,27,25)(23,30)(35,36);(2,4)(5,8,12,26)(6,14)(7,20,22,27)(9,25,10,29)(11,28,24,21)(13,30)(15,35)(16,23)(17,31)(18,32)(19,34)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,f^c=fi,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, 56501);