#123677

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

1490 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)(11,19,22,33)(12,16)(13,17)(14,18)(15,21,27,37)(20,34,36,32)(23,29)(24,30)(25,31)(26,39,28,35)(38,40);(5,6)(7,8)(9,11)(10,15,18,27)(12,20)(13,21)(14,22)(16,28,30,26)(17,19,31,33)(23,35)(24,36)(25,37)(29,34,40,32)(38,39);(2,4)(6,8)(9,12,13,23,14,24,25,38)(10,16,17,29,18,30,31,40)(11,20,21,35,22,36,37,39)(15,26,19,32,27,28,33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,b^a=bd,c^a=ce,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=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, 30873);