#8534471

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

4285 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(17,18)(19,24)(20,26)(21,31)(22,27)(23,25)(28,30)(29,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,19)(18,24)(20,25)(21,28)(22,29)(23,26)(27,32)(30,31)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(2,8)(4,15)(7,11)(10,12)(18,25)(19,26)(21,27)(29,30)(33,35,38,36)(34,40,42,39)(37,43,45,44)(41,47,48,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,20,23,24)(18,26,25,19)(21,30,27,29)(22,32,31,28)(33,36,38,35)(34,40,42,39)(37,43,45,44)(41,46,48,47);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,21,23,27)(18,22,25,31)(19,28,26,32)(20,30,24,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,22)(18,27)(19,29)(20,28)(21,25)(23,31)(24,32)(26,30)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441667);