#8824148

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	#38760	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8731344);