#219547

order	512
nilpotency class	3
isoclinism	isoclinic class of #219542
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

567 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,d^b=di,f^b=fh,d^c=di,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, 126743);