#219640

order	512
nilpotency class	3
isoclinism	isoclinic class of #219638
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	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

546 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=ef,d^2=e,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dhi,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, 126836);