#219709

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×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ehi,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, 126905);