#123523

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2538 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)(11,19,21,15)(12,23,24,16)(13,18)(14,17)(20,27)(22,32,35,28)(25,36,37,29)(26,30)(31,34)(33,38)(39,40);(2,4)(5,6)(7,8)(9,11,13,22,14,21,26,35)(10,15,18,28,17,19,30,32)(12,20,25,33,24,34,37,40)(16,27,29,38,23,31,36,39);(6,8)(9,12)(10,16)(11,20)(13,25)(14,24)(15,27)(17,23)(18,29)(19,31)(21,34)(22,33)(26,37)(28,38)(30,36)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi,g^c=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, 30719);