#8700368

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

4079 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607564);