#8734251

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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,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, 8641447);