#8195961

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	#39148	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4414 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8103157);