#8755036

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

4125 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662232);