#8417230

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
prime radical	#52743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(17,18)(19,20)(21,24)(22,29)(23,25)(26,31)(27,32)(28,30)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(2,7)(4,13)(8,16)(10,14)(17,19,24,23)(18,25,21,20)(22,27,31,30)(26,32,29,28)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,7)(5,15)(8,16)(9,12)(17,20)(18,23)(19,21)(22,28)(24,25)(26,30)(27,29)(31,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3)(2,8)(4,14)(5,9)(6,11)(7,16)(10,13)(12,15)(17,21,24,18)(19,25,23,20)(22,29,31,26)(27,32,30,28)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,15,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32);(1,5,6,15)(2,10,7,14)(3,9,11,12)(4,8,13,16)(17,23,24,19)(18,25,21,20)(22,30,31,27)(26,32,29,28)(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=gh,d^2=i,e^2=gi,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, 8324426);