#8534599

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

4287 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)(3,9)(4,10)(5,8)(6,7)(11,16)(12,13)(14,15)(17,18)(19,24)(20,26)(21,30)(22,27)(23,25)(28,32)(29,31)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(2,7)(3,10)(8,15)(11,13)(17,19,23,21)(18,24,25,30)(20,28,29,27)(22,26,32,31)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(2,7)(4,9)(8,15)(12,16)(18,25)(19,21)(26,31)(27,28)(33,35,38,36)(34,40,42,39)(37,43,45,44)(41,47,48,46);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,20,23,29)(18,26,25,31)(19,27,21,28)(22,30,32,24)(33,36,38,35)(34,40,42,39)(37,43,45,44)(41,46,48,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,23,19)(18,24,25,30)(20,28,29,27)(22,31,32,26)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,14)(2,8,7,15)(3,11,10,13)(4,12,9,16)(17,22)(18,27)(19,26)(20,24)(21,31)(23,32)(25,28)(29,30)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)

Polycyclic presentation:

2^9|b^2=gh,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, 8441795);