#10047078

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9954274);