#483226

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cgi,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	#72285888, #72285942, #72285993
maximal normal subgroup of	#72285888, #72285942, #72285993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390422);