#148379

order	512
nilpotency class	4
isoclinism	isoclinic class of #140781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,g^b=gi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi,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, 55575);