#8211630

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	#44137	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=ghi,e^2=gh,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

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, 8118826);