#496287

order	512
nilpotency class	4
isoclinism	isoclinic class of #496285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21758
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60461	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

2^9|a^2=g,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^b=gi,f^c=fh,h^c=hi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403483);