#10159201

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10066397);