#8393670

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	#40489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4476 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=ghi,e^2=h,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, 8300866);