#8934262

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	#51917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841458);