#8339845

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 8247041);