#8371531

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

4115 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8278727);