#8906978

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	#50097	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8814174);