#8987076

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	#50748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8894272);