#8824368

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4135 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gi,e^2=ghi,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, 8731564);