#219608

order	512
nilpotency class	3
isoclinism	isoclinic class of #219600
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

534 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=ef,d^2=ehi,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=di,f^b=fh,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, 126804);