#8534457

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

4305 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,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, 8441653);