#119808

order	512
nilpotency class	3
isoclinism	isoclinic class of #119807
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

517 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,e^c=eh,f^c=fi,g^c=gi

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, 27004);