#214702

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29,20,24)(21,34,36,26)(22,28)(23,27)(25,33,39,31)(30,38)(32,43)(35,47,48,42)(37,40)(41,45)(44,46);(3,9)(6,12)(10,15)(13,16)(17,19,23,20)(18,24,27,29)(21,30,36,32)(22,33,37,31)(25,40,39,28)(26,38,34,43)(35,45,48,44)(41,47,46,42);(1,3,5,9)(2,6,8,12)(4,10,11,15)(7,13,14,16)(17,20,23,19)(18,25,27,39)(21,32,36,30)(22,31,37,33)(24,28,29,40)(26,41,34,46)(35,44,48,45)(38,42,43,47);(3,9)(6,12)(10,15)(13,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,36)(24,38)(25,41)(27,34)(28,42)(29,43)(31,44)(33,45)(37,48)(39,46)(40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,f^b=fh,g^b=gh,d^c=di,f^c=fh,g^c=gh,g^d=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121898);