#123470

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1774 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,d^b=di

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, 30666);