#118188

order	512
nilpotency class	3
isoclinism	isoclinic class of #118183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

560 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:

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

Polycyclic presentation:

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

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, 25384);