#512653

order	512
nilpotency class	5
isoclinism	isoclinic class of #512651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#60830	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3498 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

833 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)(3,13,16,30)(4,9)(5,19)(6,11,22,26)(7,12)(8,15,24,31)(10,21,27,18)(14,29,17,23)(20,28,32,25);(1,3)(2,8)(4,15)(5,20)(6,16)(7,14)(9,23)(10,13)(11,28)(12,25)(17,22)(18,31)(19,30)(21,32)(24,26)(27,29);(3,14)(4,5)(9,19)(10,27)(13,29)(16,17)(18,21)(23,30);(1,4,7,5)(2,9,12,19)(3,15,14,20)(6,18,22,21)(8,23,25,30)(10,26,27,11)(13,24,29,28)(16,31,17,32)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419849);