#8339847

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	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8247043);