#8339858

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4477 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,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, 8247054);