#10159191

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3776 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)(19,29)(20,26)(21,22)(23,27)(24,28)(25,32)(30,31)(33,34,39,43)(35,36,44,40)(37,45,47,41)(38,42,46,48);(1,3,6,9)(2,5,10,15)(4,12,14,11)(7,16,8,13)(17,19,24,32)(18,25,28,29)(20,30,21,23)(22,31,26,27)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48);(1,4,6,14)(2,7,10,8)(3,11,9,12)(5,13,15,16)(17,20,24,21)(18,26,28,22)(19,30,32,23)(25,27,29,31)(33,36,39,40)(34,35,43,44)(37,46,47,38)(41,48,45,42);(1,4)(2,8)(3,12)(5,13)(6,14)(7,10)(9,11)(15,16)(17,21,24,20)(18,26,28,22)(19,23,32,30)(25,27,29,31)(33,37,39,47)(34,41,43,45)(35,42,44,48)(36,38,40,46);(1,3)(2,5)(4,11)(6,9)(7,13)(8,16)(10,15)(12,14)(17,22,24,26)(18,21,28,20)(19,31,32,27)(23,29,30,25)(33,38)(34,42)(35,41)(36,37)(39,46)(40,47)(43,48)(44,45);(1,5)(2,9)(3,10)(4,13)(6,15)(7,12)(8,11)(14,16)(17,23)(18,27)(19,20)(21,32)(22,25)(24,30)(26,29)(28,31)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=ghi,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, 10066387);