#8517583

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

4112 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,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, 8424779);