#8161278

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

4064 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8068474);