#8757167

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

4039 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=hi,f^2=gh,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, 8664363);