#8118869

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

4074 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(21,29)(24,31)(27,28)(30,32)(34,40)(36,38)(41,48)(44,46);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,46)(47,48);(2,5)(8,10)(9,12)(15,16)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,35,39,42)(34,36,40,38)(37,45,47,43)(41,46,48,44);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36,39,38)(34,35,40,42)(37,44,47,46)(41,43,48,45);(1,5,7,2)(3,10,6,8)(4,12,14,9)(11,16,13,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,37,39,47)(34,41,40,48)(35,43,42,45)(36,44,38,46);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,36)(34,42,40,35)(37,46,47,44)(41,45,48,43)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026065);