#9851060

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18,22,26)(19,21,25,23)(20,24,30,32)(27,31,28,29)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,31,32,29)(33,35,39,37)(34,40,38,42)(36,43,46,45)(41,47,44,48);(2,7)(3,9)(10,13)(11,16)(33,35,39,37)(34,40,38,42)(36,45,46,43)(41,48,44,47);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,31,23,29)(33,37,39,35)(34,40,38,42)(36,45,46,43)(41,47,44,48);(1,5)(2,10)(3,11)(4,12)(6,15)(7,13)(8,14)(9,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,32,28)(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=g,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9758256);