#483077

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
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	#3058	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

723 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390273);