#8733973

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

4621 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=gh,f^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, 8641169);