#148927

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(9,10,14,20,15,21,29,37,16,22,28,36,13,19,12,18)(11,23,24,35,25,34,27,39,26,38,31,40,30,17,32,33);(1,2,4,3)(5,6)(7,8)(9,11)(10,17)(12,27)(13,30)(14,31)(15,25)(16,26)(18,35)(19,38)(20,39)(21,23)(22,34)(24,28)(29,32)(33,36)(37,40);(1,3,4,2)(6,8)(9,12,13,28,16,29,15,14)(10,18,19,36,22,37,21,20)(11,24,25,27,26,31,30,32)(17,33,23,35,34,39,38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56123);