#8402641

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

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:

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

Polycyclic presentation:

2^9|a^2=g,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=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8309837);