#8418438

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	#44925	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=g,e^2=g,f^2=gi,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, 8325634);