#149262

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,d^c=dh,f^c=fh,h^c=hi

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, 56458);