#8533522

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

4151 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,e^2=h,f^2=gh,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, 8440718);