#117822

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

548 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:

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

Polycyclic presentation:

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

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, 25018);