#8416734

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	#52363	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gi,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, 8323930);