#480129

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#99
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1459
prime radical	C₂×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

979 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387325);