#8407194

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

3988 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(18,21)(24,26)(25,29)(31,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(3,12)(6,15)(8,16)(10,13)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,22,23,30)(18,25,21,29)(19,27,20,28)(24,31,26,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314390);