#482620

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#36019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^d=fi,f^e=fi

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, 389816);