#8406892

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=g,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, 8314088);