#495944

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#481
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,d^2=g,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,h^c=hi

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, 403140);