#8992919

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	#48910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(5,15,16,10)(17,18,22,27)(19,23,30,32)(20,25,29,24)(21,26,28,31)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(2,7)(4,14)(8,11)(10,15)(17,19,22,30)(18,23,27,32)(20,28,29,21)(24,26,25,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,7)(3,12)(4,14)(5,16)(17,20,22,29)(18,24,27,25)(19,21,30,28)(23,31,32,26)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,27)(26,32)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,21,22,28)(18,26,27,31)(19,29,30,20)(23,24,32,25)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,31)(30,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=hi,e^2=gh,f^2=g,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, 8900115);