#219660

order	512
nilpotency class	3
isoclinism	isoclinic class of #219576
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

485 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)(3,8,12,16,13,9,4,14)(17,18,22,29,23,28,40,27)(19,30,25,24,32,20,33,35)(21,36,38,26,39,45,48,44)(31,43,42,46,37,41,47,34);(1,3,6,12,7,13,5,4)(2,8,10,16,11,9,15,14)(17,19,23,32)(18,24)(20,29)(21,37,39,31)(22,25,40,33)(26,43)(27,30)(28,35)(34,36)(38,47,48,42)(41,44)(45,46);(1,4,5,13,7,12,6,3)(2,9,10,14,11,8,15,16)(17,20,22,24,23,30,40,35)(18,25,29,19,28,33,27,32)(21,34,38,41,39,46,48,43)(26,42,45,31,44,47,36,37);(17,21,22,38,23,39,40,48)(18,26,27,36,28,44,29,45)(19,31,25,42,32,37,33,47)(20,34,24,41,30,46,35,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,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=dh,f^b=fh,g^c=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, 126856);