#10040455

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	#55246	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,32)(28,31)(29,30)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(2,6)(3,9)(8,11)(10,12)(17,19,22,27)(18,23,24,32)(20,30,26,28)(21,29,25,31)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(2,6)(3,9)(8,11)(10,12)(18,24)(19,27)(20,26)(29,31)(34,41)(35,46)(36,43)(44,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,29,32,31)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19)(18,23)(20,28)(21,31)(22,27)(24,32)(25,29)(26,30)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21,22,25)(18,26,24,20)(19,29,27,31)(23,28,32,30)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 9947651);