#8119247

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	#56302	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026443);