#219679

order	512
nilpotency class	3
isoclinism	isoclinic class of #219614
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

475 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126875);