#147979

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1797
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

702 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^c=ei,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, 55175);