#148455

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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)(5,6,10,16,9,15,18,35,11,14,23,28,27,34,36,32)(7,17,8,22,19,30,24,33,21,12,26,31,20,29,25,13);(5,7)(6,12)(8,23)(9,19)(10,26)(11,21)(13,32)(14,17)(15,29)(16,22)(18,25)(20,27)(24,36)(28,31)(30,34)(33,35);(2,4)(5,8)(6,13,14,33)(7,18)(9,24)(10,20)(11,26)(12,28,17,16)(15,22,34,31)(19,23)(21,36)(25,27)(29,32,30,35)

Polycyclic presentation:

2^9|a^2=g,c^2=d,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,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55651);