#118664

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ehi,f^c=fh

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, 25860);