#8535305

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

4255 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)(3,10,8,4)(5,9,14,15)(11,16,13,12)(18,22)(23,29)(24,27)(30,32)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(2,7)(3,8)(9,15)(11,13)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(35,36)(39,42)(43,44)(46,47);(2,7)(4,10)(9,15)(12,16)(19,25)(23,29)(26,31)(30,32)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,48,47);(1,3,6,8)(2,4,7,10)(5,13,14,11)(9,16,15,12)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,36,38,35)(34,39,41,42)(37,43,45,44)(40,47,48,46);(1,4)(2,8)(3,7)(5,12)(6,10)(9,13)(11,15)(14,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,46)(45,48);(1,5,6,14)(2,9,7,15)(3,11,8,13)(4,12,10,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=ghi,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, 8442501);