#8358765

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

4743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8265961);