#219707

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,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=cei,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, 126903);