#8741965

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

4369 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8649161);