#10069257

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	#44325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9976453);