#123915

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#399
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1654 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)(9,10,14,18,15,20,13,19)(11,21,25,31,24,32,23,16)(12,17,27,35,29,36,28,34)(22,33,38,30,39,37,26,40);(2,4)(5,6)(7,8)(9,11)(10,16,20,31)(12,26,29,38)(13,23)(14,25)(15,24)(17,33)(18,21,19,32)(22,28,39,27)(30,35)(34,40)(36,37);(6,8)(9,12)(10,17)(11,22,24,39)(13,28)(14,27)(15,29)(16,30,31,40)(18,35)(19,34)(20,36)(21,37,32,33)(23,26,25,38)

Polycyclic presentation:

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

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, 31111);