#8359863

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,26)(20,21)(22,24)(23,29)(25,31)(27,28)(30,32)(33,34,39,36)(35,40,42,38)(37,41,47,44)(43,48,45,46);(2,6)(7,9)(8,13)(14,15)(17,19,22,29)(18,23,24,26)(20,30,31,27)(21,28,25,32)(33,35,39,42)(34,40,36,38)(37,45,47,43)(41,46,44,48);(4,12)(8,13)(10,16)(14,15)(18,24)(21,25)(23,26)(28,32)(35,42)(38,40)(43,45)(46,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,26)(25,32)(27,31)(33,36,39,34)(35,38,42,40)(37,41,47,44)(43,48,45,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,25)(19,27)(21,24)(22,31)(23,28)(26,32)(29,30)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,25)(18,20,24,31)(19,28,29,32)(23,30,26,27)(33,38)(34,35)(36,42)(37,46)(39,40)(41,43)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8267059);