#121469

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#1444
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

462 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)(3,7)(4,6)(5,12)(8,11)(9,15)(10,14)(13,16)(17,18,22,26,23,28,21,27)(19,29,32,40,33,43,31,24)(20,34,37,42,38,44,36,25)(30,46,47,41,48,39,35,45);(2,6)(3,9,10,13)(7,15,14,16)(8,12)(17,19,21,31,23,33,22,32)(18,24,26,29,28,40,27,43)(20,35,36,48,38,47,37,30)(25,41,34,39,42,45,44,46);(1,3)(2,7)(4,10)(5,13)(6,14)(8,15)(9,11)(12,16)(17,20,21,36,23,38,22,37)(18,25,26,34,28,42,27,44)(19,30,31,35,33,48,32,47)(24,39,29,45,40,46,43,41)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28665);