#120990

order	512
nilpotency class	3
isoclinism	isoclinic class of #120987
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

383 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)(3,8,4,14)(5,12)(6,10)(7,11)(9,15,16,13)(17,18,22,28,23,27,21,26)(19,29,31,39,32,41,36,24)(20,33,34,42,35,25,37,40)(30,44,46,48,47,43,45,38);(1,3,6,13,5,4,7,15)(2,8,10,9,12,14,11,16)(17,19)(18,24,27,39)(20,30)(21,31)(22,36)(23,32)(25,38,33,48)(26,29,28,41)(34,45)(35,47)(37,46)(40,43,42,44);(1,4,5,3)(2,9)(6,15,7,13)(8,11)(10,14)(12,16)(17,20,21,34,23,35,22,37)(18,25,26,40,27,33,28,42)(19,30,31,45,32,47,36,46)(24,38,29,43,39,48,41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28186);