#10159218

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	#50965	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3927 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)(19,28)(20,26)(21,22)(23,25)(24,27)(29,31)(30,32)(33,34,40,45)(35,36,47,41)(37,48,46,42)(38,43,44,39);(1,3,6,9)(2,5,10,15)(4,12,14,11)(7,16,8,13)(17,19,24,23)(18,25,27,28)(20,29,21,30)(22,31,26,32)(33,35)(34,41)(36,45)(37,43)(38,48)(39,46)(40,47)(42,44);(1,4,6,14)(2,7,10,8)(3,11,9,12)(5,13,15,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,32,28,31)(33,36,40,41)(34,35,45,47)(37,44,46,38)(39,48,43,42);(1,4)(2,8)(3,12)(5,13)(6,14)(7,10)(9,11)(15,16)(17,21,24,20)(18,26,27,22)(19,30,23,29)(25,32,28,31)(33,37,40,46)(34,42,45,48)(35,43,47,39)(36,38,41,44);(1,3)(2,5)(4,11)(6,9)(7,13)(8,16)(10,15)(12,14)(17,22,24,26)(18,21,27,20)(19,31,23,32)(25,30,28,29)(33,38)(34,43)(35,42)(36,37)(39,45)(40,44)(41,46)(47,48);(1,5)(2,9)(3,10)(4,13)(6,15)(7,12)(8,11)(14,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,39,40,43)(34,44,45,38)(35,46,47,37)(36,48,41,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=ghi,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, 10066414);