#8824356

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	#50768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731552);