#8975459

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

4043 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34)(35,40)(36,42)(37,43)(38,48)(39,41)(44,47)(45,46);(2,6)(7,9)(8,12)(14,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,39,38)(34,40,41,48)(36,44,46,43)(37,42,47,45);(2,6)(3,10)(4,13)(14,15)(19,26)(20,24)(21,29)(23,30)(34,41)(35,38)(36,46)(37,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36)(34,42)(35,44)(37,48)(38,43)(39,46)(40,47)(41,45);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37,39,47)(34,43,41,44)(35,45,38,42)(36,40,46,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38,39,35)(34,40,41,48)(36,43,46,44)(37,42,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=gh,f^2=h,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, 8882655);