#123653

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,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, 30849);