#8258568

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

4180 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:

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

Polycyclic presentation:

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

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, 8165764);