#10166242

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	#53739	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18,22,24)(19,20,27,23)(21,25,31,32)(26,28,29,30)(33,34)(35,38)(36,41)(37,46)(39,42)(40,45)(43,44)(47,48);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,48)(46,47);(2,9)(3,4)(10,14)(11,12)(18,24)(20,23)(25,32)(28,30)(33,36,39,46)(34,41,42,37)(35,43,45,47)(38,44,40,48);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,20,22,23)(18,19,24,27)(21,30,31,28)(25,29,32,26)(33,37)(34,36)(35,44)(38,47)(39,41)(40,43)(42,46)(45,48);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,21,22,31)(18,25,24,32)(19,26,27,29)(20,28,23,30)(33,34)(35,40)(36,41)(37,46)(38,45)(39,42)(43,48)(44,47);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,38,39,40)(34,35,42,45)(36,44,46,48)(37,47,41,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 10073438);