#119012

order	512
nilpotency class	3
isoclinism	isoclinic class of #119009
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26208);