#8399490

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	#39503	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306686);