#8963080

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	#49825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870276);