#8621368

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528564);