#493664

order	512
nilpotency class	4
isoclinism	isoclinic class of #493657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#63037	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2184 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400860);