#10166322

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	#57197	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=h,e^2=hi,f^2=gh,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, 10073518);