#149304

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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)(3,8)(4,6)(5,7)(9,10,15,21,16,19,28,37,14,20,27,38,13,29,12,18)(11,22,25,36,24,17,31,34,26,35,40,32,30,33,23,39);(1,3,5,4)(2,6,7,8)(9,11,16,24,14,26,13,30)(10,17,19,35,20,33,29,22)(12,23,15,25,28,31,27,40)(18,36,21,34,37,32,38,39);(1,4,5,3)(2,6,7,8)(9,12,16,15,14,28,13,27)(10,18,19,21,20,37,29,38)(11,23,24,25,26,31,30,40)(17,32,35,39,33,36,22,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56500);