#8110610

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	#59220	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8017806);