#8195852

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	#59039	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4488 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=h,f^2=ghi,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, 8103048);