#483221

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285467, #72285618, #72285630
maximal normal subgroup of	#72285467, #72285618, #72285630

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390417);