#10040536

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	#46080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,6)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,18)(19,23)(20,22)(21,25)(24,27)(26,30)(28,29)(31,32)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(1,2,5,8)(3,6,10,14)(4,7,12,11)(9,13,16,15)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,4,5,12)(2,7,8,11)(3,9,10,16)(6,13,14,15)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=h,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, 9947732);