#383840

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291036);