#8517581

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

3954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=g,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, 8424777);