#10069261

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	#41140	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(35,41)(39,42)(43,47)(46,48);(2,6)(7,13)(8,14)(15,16)(18,20)(19,25)(24,28)(27,31)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(3,9)(7,13)(10,11)(15,16)(18,20)(19,25)(24,28)(27,31)(34,36)(39,42)(40,44)(46,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3)(2,7)(4,11)(5,9)(6,13)(8,16)(10,12)(14,15)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=hi,f^2=g,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, 9976457);